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© Copyright 2010 DigiPen Institute Of Technology and DigiPen (USA) Corporation. All rights reserved.
Finding Strategies to Solve a 4x4x3 3D Domineering Game
BY Jonathan Hurtado
B.A. Computer Science, New York University, 2001
THESIS
Submitted in partial fulfillment of the requirements for the degree of Master of Science
in the graduate studies program of DigiPen Institute Of Technology
Redmond, Washington United States of America
Fall 2010
Thesis Advisor: Matt Klassen
DIGIPEN INSTITUTE OF TECHNOLOGY
GRADUATE STUDY PROGRAM
DEFENSE OF THESIS
THE UNDERSIGNED VERIFY THAT THE FINAL ORAL DEFENSE OF THE
MASTER OF SCIENCE THESIS OF Jonathan Hurtado
HAS BEEN SUCCESSFULLY COMPLETED ON December 8, 2010
TITLE OF THESIS: Finding Strategies to Solve 4x4x3 3d Domineering Game
MAJOR FILED OF STUDY: COMPUTER SCIENCE.
COMMITTEE:
Dr. Matt Klassen, Chair Dr. Xin Li
Dr. Dmitri Volper Dr. Ton Boerkoel
APPROVED :
Dr. Xin Li date Dr. Xin Li date Graduate Program Director Dean of Faculty
Samir Abou-Samra date Claude Comair date Department of Computer Science President
The material presented within this document does not necessarily reflect the opinion of the Committee, the Graduate Study Program, or DigiPen Institute of Technology.
INSTITUTE OF DIGIPEN INSTITUTE OF TECHNOLOGY
PROGRAM OF MASTER’S DEGREE
THESIS APPROVAL
DATE: December 8, 2010
BASED ON THE CANDIDATE’S SUCCESSFUL ORAL DEFENSE, IT IS
RECOMMENDED THAT THE THESIS PREPARED BY
Jonathan Hurtado
ENTITLED
Finding Strategies to Solve a 4x4x3 3D Domineering Game
BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF COMPUTER SCIENCE FROM THE
PROGRAM OF MASTER’S DEGREE AT DIGIPEN INSTITUTE OF
TECHNOLOGY.
Dr. Matt Klassen Thesis Advisory Committee Chair
Dr. Xin Li Director of Graduate Study Program
Dr. Xin Li Dean of Faculty
The material presented within this document does not necessarily reflect the opinion of the Committee, the Graduate Study Program, or DigiPen Institute of Technology.
Acknowledgments
I would like to thank my thesis advisor, Matt Klassen, for his guidance, expertise,
patience, and understanding. I would not have finished my thesis without his
support and assistance.
Table of Contents
1.0 – INTRODUCTION TO COMBINATORIAL GAMES & DOMINEERING ................. 1
1.1 – COMBINATORIAL GAMES ............................................................................ 1 1.2 – THREE-PLAYER COMBINATORIAL GAMES................................................. 19
2.0 – STRATEGIES TO SOLVE A 3X3X3 DOMINEERING GAME ......................... 29
2.1 – STRATEGY TO BLOCK M IN A 3X3X3 DOMINEERING GAME ...................... 29 2.2 – ANALYSIS OF L-R-M AND R-L-M GAMES ................................................ 33 2.3 – ANALYSIS OF L-M-R AND R-M-L GAMES ................................................ 38 2.4 – ANALYSIS OF M-L-R AND M-R-L GAMES ................................................ 41
3.0 – STRATEGIES TO SOLVE A 4X4X3 DOMINEERING GAME: PART 1 ............ 54
3.1 – ANALYSIS OF L-R-M AND R-L-M GAMES ................................................ 56 3.2 – ANALYSIS OF M-L-R AND M-R-L GAMES ................................................ 61 3.3 – ANALYSIS OF L-M-R AND R-M-L GAMES ................................................ 68 3.4 – ENDGAME ANALYSIS................................................................................. 73
4.0 – STRATEGIES TO SOLVE A 4X4X3 DOMINEERING GAME: PART 2 ............ 85
4.1 – PART 1 OF ALLIANCE’S WINNING STRATEGY............................................ 85 4.2 – PART 2 OF ALLIANCE’S WINNING STRATEGY.......................................... 100 4.3 – RESULTS FROM SIMULATION USING ALLIANCE’S WINNING STRATEGY... 105
5.0 – FUTURE WORK ............................................................................... 110
REFERENCES........................................................................................... 111
1.0 – Introduction to Combinatorial Games & Domineering
In this chapter, we will introduce several topics to help the reader understand the material
covered in this thesis.
1.1 – Combinatorial Games
Combinatorial game theory is the study of combinatorial games and differs from the
traditional analysis of game theory, which reviews economic, zero-sum games. In zero-
sum games, the gains of one player are offset by the losses of the other player, which
equals the sum of zero. In these games, the outcome of how each player wins or loses is
determined by a payoff matrix, and neither player will know exactly what the other
player will do since they both have to move simultaneously. The following section
explains what a combinatorial game is and how it differs from economic games.
What is a combinatorial game?
A combinatorial game is a game where two players (often called L for left player and R
for right player) play in alternate turns until one player is unable to make any legal
moves. Usually, the player who cannot make any more legal moves loses, but there are
variations to the combinatorial game called misére where the player who cannot make
any more legal moves wins. There are no ties in a combinatorial game—there must be
one winner and one loser. Nothing about the combinatorial game is hidden from either
player as both players know all the details of the game’s state at all times (this is referred
to as perfect information). A combinatorial game has no element of chance like a dice
roll or a card deal. There are also no cycles in a combinatorial game, meaning that a
position in a combinatorial game is not repeated in a later turn. For this thesis, we will
only consider finite combinatorial games, which end after a finite sequence of moves.
If a game violates any of the above criteria, then it is not a combinatorial game. For
example, checkers is not considered a combinatorial game because it can have cycles,
meaning that positions can be repeated. Chess also cannot be classified as a
combinatorial game because the game can end in a draw. Zero-sum games, which we
1
••
mentioned in the beginning of this chapter, are not combinatorial games because moves
by both players are performed simultaneously instead of in alternate turns.
Now that we have provided examples of what does not constitute a combinatorial game,
we will show a game that does fulfill the requirements of a combinatorial game,
Domineering.
What is Domineering?
Domineering is a two-player combinatorial game that is played on a grid board of any
size (ex. 3x3, 4x6, 10x4, etc.), although it is usually square to be fair to both players.
Each player can place a domino piece that covers two adjacent spaces on the board. Both
players take turns placing a domino on the board (the domino cannot overlap other
domino pieces) until one player is unable to place any more pieces, which results in a loss
for that player. One player, referred to as left player or L, can only place pieces vertically
on the board. The other player, referred to as right player or R, can only place pieces
horizontally on the board. Diagrams 1-1a to 1-1c show a brief example of a Domineering
game. In this and subsequent examples of Domineering games, moves made in the
current turn will appear black, and moves made in previous turns will appear gray.
Example of a 3x3 Domineering Game
• ••
•••
Diagram 1-1a Diagram 1-1b Diagram 1-1c L makes his first move. R then makes his first move. L then makes his second move.
In this 3x3 Domineering game, the first player (L) starts by placing a piece vertically on
the upper-left corner of the board in Diagram 1-1a. The second player (R) places a piece
horizontally on the upper-right corner of the board in Diagram 1-1b. L places his second
piece vertically on the center of the board in Diagram 1-1c. This move prevents R from
placing any more horizontal pieces on the board, which results in a loss for R.
••
••
2
• •
• • •
• • • •
• • • •
Domineering is classified as a partizan combinatorial game.
Impartial vs. Partizan Combinatorial Games
There are two types of combinatorial games: impartial and partizan. An impartial
combinatorial game is one where, in any position, the moves available to be played by
either player are identical. A partizan combinatorial game is a game where the moves
both players make are different. To show the differences between impartial and partizan
games, we will introduce another combinatorial game called Nim. Nim is played with
heaps of counters, as demonstrated in the diagram below.
•
Diagram 1-2 An example of Nim
Each player takes turns removing any number of counters (but at least one) from one
heap until there are no more counters (a player loses when there are no more counters to
remove in his turn). Nim is an impartial game because both left and right players have
the same moves: removing any number of counters from a heap. Domineering is a
partizan game because both players have different moves: one player can only place
pieces vertically on the board, while the other can only place pieces horizontally on the
board.
It is important to note the distinction between impartial and partizan games because the
number of outcome classes is smaller for impartial games.
What is an outcome class?
An outcome class describes which player in a combinatorial game is able to win
regardless of what his opponent does (also known as “forcing a win”) provided that the
player follows a specific strategy (also known as the winning strategy). It is derived from
3
•
•
•
the “Fundamental Theorem of Combinatorial Games” introduced in Lessons in Play
[ANW_07], which is defined as:
In two-player finite combinatorial games, exactly one player always has a winning
strategy for a given turn order.
In Lessons in Play, the authors presented the proof of this theorem as follows: presume a
finite combinatorial game G where the first player can force a win going first or the
second player can force a win going second, but not both. Each of the first player’s
moves goes to a position that, by induction, is either a win for the second player going
first or the first player going second. If any of the first player’s moves belongs to the
latter category, then the first player can force a win by choosing one of them. However,
if all of the first player’s moves are in the former category, then the second player can
force a win by using his winning strategy in the position resulting from any of the first
player’s moves [ANW_07].
From the “Fundamental Theorem of Combinatorial Games”, we can define two outcome
classes for impartial combinatorial games: N, which means the first player (or next
player) can force a win, and P, which means the second player (or previous player) can
force a win. The following diagram shows an example of an N game using Nim.
Diagram 1-3 A Nim game in outcome class N
The Nim game in Diagram 1-3 is in N because the first player can always force a win by
removing all the counters from the single heap. The following shows an example of a P
game using Nim.
4
• •
• •
• •
Diagram 1-4 A Nim game in outcome class P
The Nim game in Diagram 1-4 is in P because the second player can force a win by
keeping the two heap stacks even until one heap is completely removed by the first player
(the second player can then remove what’s left from the remaining heap to win). For
example, if the first player were to remove one counter from one heap, the second player
should remove one counter from the other heap. As long as the second player follows the
winning strategy of keeping the heaps even, there is nothing that the first player can do to
win.
When the Fundamental Theorem is applied to partizan combinatorial games, there are
four outcome classes, which are defined in the following chart.
Outcome Class When Left moves first When Right moves first N Left wins Right wins
P Right wins Left wins
L Left wins Left wins
R Right wins Right wins
Diagram 1-5 Outcome classes for partizan combinatorial games
In additional to N and P, partizan games also have the outcome classes L, which means
that the left player can force a win regardless of turn order, and R, which means that the
right player can force a win regardless of turn order. Partizan combinatorial games have
more outcome classes than impartial combinatorial games because the moves that left and
right player make in partizan games are different. The following demonstrate examples
of L and R games using Domineering.
Diagram 1-6a Diagram 1-6b A Domineering game in outcome class L A Domineering game in outcome class R
5
Diagram 1-6a shows a L Domineering game where left player can win regardless of turn
order because right player is unable to move in that game. Likewise, Diagram 1-6b
shows a R Domineering game where right player can win regardless of turn order
because left player is unable to move in that game.
In the next section, we will introduce another important aspect of combinatorial games,
their game tree.
Combinatorial Game Trees
A game tree is a directed graph whose nodes represent all the game's possible positions
and whose edges represent all possible moves that can be made in the game. Typically,
the positions on the game tree’s even levels represent moves made by the first player, and
the positions on the tree’s odd levels represent moves made by the second player
[Ros_03]. The following diagram shows an example of a game tree with the first
player’s moves marked with ‘L’, and the second player’s moves marked with ‘R’.
Diagram 1-7 An example of a game tree
However, the game tree of a combinatorial game is constructed differently, in part
because of partizan combinatorial games. Recall that in a partizan combinatorial game,
the moves left and right player make are not the same. Therefore, the children of a
position in a combinatorial game tree must be partitioned into two sets, one for the left
player and one for the right player. Moves that can be made at a position by the left
player are drawn as the position’s left children and are considered to be the position’s left
options. Moves that can be made at a position by the right player are drawn as the
6
position’s right children and are considered to be the position’s right options. The
following diagram shows an example of a game tree for a combinatorial game.
Diagram 1-8 An example of a combinatorial game tree
One could consider a combinatorial game tree as a combination of two game trees (with
each tree having a different player going first). However, the reader may notice in
Diagram 1-8 that several positions in the tree are the result of two consecutive left or
right options, even though such moves are not possible in a game with alternating turns.
There are two reasons for this. The first is that both children are needed to calculate the
outcome class of a position (which we will demonstrate later in this chapter). The second
is that a game tree can represent a summand in a sum of games. Thus, it is possible that
one player can move consecutively on one summand game if the other player plays on
another summand game.
The Value of a Combinatorial Game
Now that we have introduced the combinatorial game tree, we can use them to
demonstrate the recursive definition of a combinatorial game. A combinatorial game, G,
can be defined as {GL | GR}, where GL represents the set of G’s left options (which are also
combinatorial games) and GR represents the set of G’s right options (also combinatorial
games) [ANW_07]. The base case for this recursive definition is GL = Ø and GR = Ø (Ø
represents the null set) because there exist combinatorial games where the left and right
players will not be able to make any moves. From this recursive definition, J.L. Conway
developed the simplest possible combinatorial games and defined values for each of
them, which are demonstrated in Diagram 1-11 [Con_76]. We also listed the outcome
classes of each game to show how Conway’s simple games correlate to the outcome
classes we defined earlier in this chapter.
7
G
{ | } = 0 { 0 | } = 1 { | 0 } = -1 { 0 | 0 } = *
P L R N
Diagram 1-9 Conway’s simple combinatorial games and their outcome classes
The first game in Diagram 1-9 is defined as 0 and has no left or right options (meaning GL
and GR are empty, and { Ø | Ø } can be rewritten as { | }). This game is in the P outcome
class, a win for the second player. This is because whoever goes first automatically loses,
as there are no moves available. Whenever you see the game 0 mentioned later in this
chapter, remember that it is a combinatorial game where neither left nor right player can
make a move.
The second game in Diagram 1-9 is defined as 1 and has 0 (the first game in Diagram 1-
9) as its left option and the empty set as its right option. Since the right player cannot
move in this game regardless of whether he goes first or second, the second game is a win
for the left player, so it is in outcome class L.
The third game in Diagram 1-9 is defined as -1 and has the empty set as its left option
and 0 (the first game in Diagram 1-9) as its right option. Since the left player cannot
move in this game regardless of whether he goes first or second, the third game is a win
for the right player, so it is in outcome class R.
The last game in Diagram 1-9 is defined as * and has 0 as its left and right options. It is
in the N outcome class, a win for the first player, because after the first player moves, the
second player cannot move and therefore loses.
Canonical Form of Combinatorial Games
An interesting property of combinatorial games that can make traversing a combinatorial
game tree easier is that they can be reduced to a simpler game called the canonical form.
8
The canonical form of a game G is equivalent to G in the sense that it behaves the same
as G in any sum of games—that is the outcome of the sum is preserved (see page 78 of
[ANW_07]). This form can be reached by removing the dominated and reversible
options of a game, which we will explain in this section.
To demonstrate what a dominated option is in a combinatorial game, we will use the
following Domineering game in Diagram 1-10 as an example.
Diagram 1-10
If left player were to move first, he has two possible moves in this game, as demonstrated
in Diagrams 1-11a and 1-11b as games H1 and H2.
•• •
•
Diagram 1-11a Diagram 1-11b Game H1 Game H2
The move made in Diagram 1-11a (Game H1) is great for the left player because the right
player is unable to make a move, making it a win for the left player. The move made in
Diagram 1-11b (Game H2), on the other hand, is bad for the left player because he loses,
as the right player covers the remaining spaces of the game with his next move. The
second move is considered dominated by the first move because it is worse for the left
player. Technically, H1 ≥ H2, or L wins playing second on H1 – H2, which is equivalent
to H1 – H2 œ L » P [ANW_07]. When simplifying the game in Diagram 1-10 to its
canonical form, the dominated option can be removed from the game tree.
To demonstrate what a reversible option is in a combinatorial game, we will introduce the
combinatorial game, up, which is defined as the following [ANW_07]:
9
• = { 0 | * }
Diagram 1-12
This game is in the L outcome class because the left player can force a win regardless of
whether he goes first or second. The left option is 0 (see the first game in Diagram 1-9),
which is a game where the right player cannot move in his turn, resulting in the right
player’s loss. The right option is * (see the fourth game in Diagram 1-9), which is a win
for the left player because * is an N game and the first player is the next player to move.
Next, we will construct a partial game tree for the game, •*.
Diagram 1-13
•* also means •+*, so both players can either play on • or * in their first turn. The
positions from those possible moves are listed as the children in the Diagram 1-13 game
tree. See the fourth game in Diagram 1-9 and the definition in Diagram 1-12 to review
what happens when the left and right players play on * and •. The reader will notice that
none of the child positions are listed as * + 0 or • + 0, even though some of the results of
playing on • or * is 0 (although when right player plays on •, it is *). Since 0 is a game
in which neither player can make any more moves, we can omit that game from the game
tree if it is a summand with another game.
One of the possible positions from •* for the right player is * + * (if he played on • in
his first turn). We simplified * + * to 0 because when the first player moves on * + *, the
remaining position is * + 0, or *. When the second player moves on *, the resulting
position is 0.
10
The game’s right option that leads to • is dominated by the right option that leads to 0
because it is a worse move for right player. Recall that • is in L, or a win for the left
player. Therefore, the right player is better off playing to the position that goes to 0, as
that results in a win for the right player (since left player cannot move in the 0 game).
We will remove the dominated option in subsequent game tree diagrams. The reader
may notice in Diagram 1-13’s left options that * is a worse move for left player than •,
but because •-* is an N game, * cannot be removed as a dominated option.
A reversible option is defined as an option that an opposing player can immediately
respond to such that it leaves the opposing player in as good or better position than before
the reversible option is played. We will continue our •* game tree example to
demonstrate this option.
Diagram 1-14
Note that after right player moves on •, the position is *. After left plays on *, it is a loss
for the right player because the right player cannot move in the 0 game. Therefore, the
second level • is a reversible option because its right option leaves the left player at a
better position than •*. A reversible option and its subsequent option are collapsed into
the game tree, which in this case is the second level • and its right child, *.
11
Diagram 1-15
There are no more dominated or reversible options, so we have reached the canonical
form of •*, which is { 0, * | 0 }. For a proof that this simplified game is equivalent to the
original game { *, • | 0, • }, see Chapter 4 of [ANW_07].
The First Solved Combinatorial Game
A combinatorial game is considered solved if one can find which player has a winning
strategy for each turn order. The first combinatorial game to be solved was Nim, back in
1901 by Charles L. Bouton [Bou_02]. He presented a mathematical proof that stated the
player’s winning strategy for each turn order based on the number of counters and heaps
in a Nim game. To present this proof, we first need to introduce nim-sum. Nim-sum is
the binary digital sum of numbers that discards any carries from one digit to the next
[ANW_07]. In other words, it is the “bit-wise” sum of the digits modulo 2. The nim-
sum operand is ∆ to distinguish it from +. The following shows an example of
calculating 5 ∆ 6 ∆ 7.
5 1 0 1
6 1 1 0
7 1 1 1
1 0 0
Diagram 1-16 A demonstration to calculate 5 ∆ 6 ∆ 7
The answer to the example in Diagram 1-16 is 100, which is binary for 4. For each
column of binary digits, if there is an even number of 1s, then the column sum is 0;
otherwise, the column sum is 1.
12
• •
• • •
• • •
• • •
• • •
• • •
Earlier in this chapter, we showed the winning strategies for Nim when the number of
heaps was 1 (that game is in outcome class N) or 2 (that game is in outcome class P).
For all Nim games with heap size a, b, …, k, the games are in the outcome class P if a ∆
b ∆ … ∆ k = 0; otherwise, they are in outcome class N [ANW_07]. We will demonstrate
this using the following Nim game as an example.
•
Diagram 1-17 A Nim game with heap sizes 5, 6, and 7
We already calculated the nim-sum of 5 ∆ 6 ∆ 7 to be 4. Since the nim-sum is not 0, this
game is in outcome class N. The first player’s winning strategy is to remove the correct
number of counters from a heap so that the game’s new nim-sum is 0. This can be
achieved by nim-adding the game’s current nim-sum (which in this case is 4) and the
number of counters in one heap. If this new nim-sum is less than the current size of the
heap, the first player can reduce the heap’s size to create a 0 nim-sum game. Let’s apply
this strategy to the Diagram 1-17 game.
The Diagram 1-17 game has heap sizes 5, 6, and 7. If you nim-added those three values
to the game’s nim-sum (which is 4), you get 1, 2, and 3 respectively (5 ∆ 4 = 1, 6 ∆ 4 =
2, and 7 ∆ 4 = 3). Since 1 < 5, 2 < 6, and 3 < 7, the first player can reduce any of the
three heaps to those values to create a 0 nim-sum game, which is demonstrated in the
following diagrams.
13
0
1 0 0 1
6 1 1 0
7 1 1 1
0 0 0
Diagram 1-18a 1 ∆ 6 ∆ 7 = 0
5 1 0 1 5 1 0 1
2 0 1 0 6 1 1 0
7 1 1 1 3 0 1 1
0 0 0
Diagram 1-18b 5 ∆ 2 ∆ 7 = 0
0 0
Diagram 1-18c 5 ∆ 6 ∆ 3 = 0
In Diagram 1-18a, the first player reduced the heap of size 5 to 1. In Diagram 1-18b, the
first player reduced the heap of size 6 to 2. In Diagram 1-18c, the first player reduced the
heap of size 7 to 3. Note that the first player only needs to reduce one heap size to create
a 0 nim-sum game.
No matter which and how many counters the second player removes in his next turn, the
first player will always win as long as he follows the winning strategy of removing the
correct number of counters to make the game’s nim-sum 0.
A proof of this winning strategy can be found in pages 138-140 in [ANW_07]. For many
other combinatorial games, a general solution is not common, but specific instances can
be solved with a game tree analysis. In the next section, we will demonstrate how to use
a game tree to solve a combinatorial game.
Solving a Combinatorial Game
One method to solve a combinatorial game is to construct its game tree. From that game
tree, one can calculate the game’s outcome class and determine whether a winning
strategy for each turn order exists. This process is similar to the typical method of
analyzing a game tree, which is to calculate the value of each tree’s node and use a
minimax strategy to pick the best moves for each player [Ros_03].
The minimax strategy is a strategy where the first player picks a position whose child has
the maximum value and the second player picks a position whose child has the minimum
value. The following diagram shows an example of a minimax tree.
14
Diagram 1-19 An example of a minimax game tree
Note in Diagram 1-19 how the max player (whose nodes are squares) picked the
maximum value from his available moves and the min player (whose nodes are ellipses)
picked the minimum value from each of his available moves. Combinatorial games must
have one winner and one loser, so the values can be restricted to 1 if it is a win for the
first player and -1 if it is a win for the second player. The following chart demonstrates
how those values correlate to the outcome classes we defined earlier in this chapter.
Outcome Class When Left moves first When Right moves first N Left wins : 1 Right wins : 1
P Right wins : -1 Left wins : -1
L Left wins : 1 Left wins : -1
R Right wins : -1 Right wins : 1
Diagram 1-20
Once the outcome classes for each position of the game tree is calculated, the minimax
strategy can be used to determine the player’s best moves for each turn order in a
combinatorial game. The next section demonstrates an example of this process using a
Domineering game tree.
Solving a Domineering Game
A Domineering game is considered solved if one can find which player has a winning
strategy for each turn order. We will show an example of how to solve a Domineering
game by constructing a game tree for the Domineering game in Diagram 1-20.
15
Diagram 1-21
There are a few things to be aware about constructing a combinatorial game tree. The
first is that symmetric moves are only included once in a game tree. Diagrams 1-22a and
1-22b show an example of symmetric moves.
••
••
Diagram 1-22a Diagram 1-22b
After L places his vertical piece in both Diagrams 1-22a and 1-22b, one can see that the
boards’ empty spaces are equal when you account for symmetry.
The second is that any covered squares or isolated squares (with no other side-adjacent
squares) that neither player can play are ignored. Thus, the result of Diagrams 1-22a and
1-22b can be written as follows:
Diagram 1-23
The exception to this rule is when the game ends. If the result of a position is just an
isolated square (or several separate isolated squares), then one isolated square can be
listed as that position’s child. Let’s construct the entire game tree for the game in
Diagram 1-21.
16
Diagram 1-24
Once the game tree is constructed, the next goal to solving a Domineering game is to find
the outcome class of the game tree’s root position. The first step of this process is to
calculate the outcome class of the tree’s leaves. In Diagram 1-24, all the leaves are single
isolated games (or 0 games), and those games are P because the next player is unable to
place a piece in his turn, resulting in a loss for that player. So let’s replace those leaves
with P.
Diagram 1-25
The next step is to calculate the outcome classes of the positions above the leaves. The
following chart in Diagram 1-26, defined in Lessons in Play, explains how to calculate
the outcome class of a position based on the outcome classes of its children [ANW_07].
How to Determine an Outcome Class for a Combinatorial Game Position
some right option œ R » P all right options œ L » N
some left option œ L » P N L
all left options œ R » N R P
Diagram 1-26
17
For example, the chart says that if at least one left option and at least one right option of a
position are P, then the position itself is N, so we’ll replace the positions above the two P
options in Diagram 1-25 with N. Diagram 1-25 also has a three-space column game with
no right children, which means that the game’s set of right options is the empty set. The
outcome class for that position is L since it has one left option in L and the “all right
options œ L » N” is true because there are no right options.
Diagram 1-27
This process is repeated as you move up the game tree until you ultimately calculate the
outcome class of the game tree’s root position. Let’s analyze the root position in
Diagram 1-25 using the chart from Diagram 1-26.
Diagram 1-28
Diagram 1-28 shows that after calculating every position’s outcome class, the root
position’s outcome class is L. This means that left player can force a win regardless of
whoever goes first. The reader can see in Diagram 1-26 which moves would guarantee a
victory for left player for both turn orders (those moves are left player’s winning
strategy). Thus, the Domineering game in Diagram 1-21 is solved.
18
Solved Domineering Games
The larger a Domineering board is, the more branches its game tree will have, making it
harder to determine the game’s outcome class by hand. Artificial intelligence researchers
have looked into solving the larger boards by constructing search engines to calculate
their outcome class. These search engines use a combination of alpha-beta pruning and a
move ordering heuristic to remove as many unnecessary branches as possible from the
engine’s traversal and analysis of the game tree [BUH_00, Bul_02].
Brueker, Uiterwijk and van den Herik created the program DOMI to solve m x n boards
where 2 ≤ m ≤ 8 and m ≤ n ≤ 9 [BUH_00]. Bullock later improved on their research by
developing a search application called Obsequi that could not only solve those games
faster, but also solve a 10 x 10 board, which is currently the largest solved Domineering
game [Bul_02].
Two-player combinatorial games and 2D Domineering have been researched extensively
by mathematicians and computer scientists. The three-player variants of those games, on
the other hand, have not been explored as much. In the next section, we will introduce
three-player combinatorial games, specifically 3D Domineering, and the steps needed to
solve a 3D Domineering game.
1.2 – Three-Player Combinatorial Games
We defined combinatorial games in the previous section as two-player games, but there
exist variants of those games that can be played with three players. These three-player
games can still be considered combinatorial games as long as they have the following
properties: alternating turns, no ties, loss for player with no more legal moves, perfect
information, no elements of chance, and no cycles.
The addition of the third player makes analyzing combinatorial games more complex and
complicated. For example, to construct a game tree for a three-player combinatorial
game, the children of each position must be divided into three distinct sets, one for the
left player, one for the middle player, and one for the right player (as opposed to two
distinct sets for two-player trees). Options for the left, middle, and right player are drawn
19
as a position’s left, middle, and right children respectively. The following diagram is an
example of a game tree for a three-player combinatorial game, with the first, middle, and
right options marked as ‘L’, ‘M’, and ‘R’ respectively.
Diagram 1-29 An example of a three-player combinatorial game tree
Outcome Classes for Three-Player Games
Another aspect of combinatorial games that changes with the addition of a third player is
their outcome classes. The outcomes classes that were derived from the Fundamental
Theorem for two-player combinatorial games (where exactly one player has a winning
strategy for each turn order) do not apply to three-player combinatorial games. This is
because it is possible in a three-player game that no player can have a winning strategy in
a turn order. Philip Straffin explored this possibility through a concept introduced in his
paper called “decision by player” [Str_85]. A “decision by player” means that one player
who is unable to win a three-player impartial game can affect which of the other two
players can win with his next move. Thus, it is possible that one player cannot force a
win no matter what he does if one of his opponents plays a move that causes the other
opponent to win.
James Propp defined the scenario where no player can force a win as “queer” or for
three-player impartial games [Pro_00]. Alessandro Cincotti also used the term to
describe the same scenario for three-player partizan games [Cin_08]. We suggest that we
change this term to “Qual,” which still maintains the moniker that fits with the other
outcome classes defined for three-player games (such as N, O, and P, which we’ll
introduce a bit later). Qual is German for “torment” and is used in the German saying,
“Wer die Wahl hat, hat die Qual,” which translates to, “Whoever has the choice, has the
torment.” This saying seems appropriate for games where no one can force a win as
20
• •
• •
whoever has the first choice in a game also has the torment of not having a winning
strategy. The following is an example of a game using Nim.
•
Diagram 1-30 A Nim game in outcome class
The Nim game in Diagram 1-30 is because none of the three players can force a win.
The first player cannot win because he cannot remove both heaps in his first turn, and
what’s left will be taken by the other two players before the first player’s next turn. The
second player cannot win if the first player removes one counter from the two-counter
heap, leaving the second player with two heaps. The second player can only remove one
heap in his next turn, leaving the other heap for the third player to remove and win. The
third player cannot win if the first player removes one heap stack from the game and the
second player removes the other heap stack.
James Propp also defined the following outcome classes for three-player impartial games:
N means that the first player (or next player) can force a win. O means that the second
player (or other player) can force a win. P means that the third player (or previous
player) can force a win. The following Nim games show examples of these outcome
classes.
•
Diagram 1-31 A Nim game in outcome class N
The Nim game in Diagram 1-31 is N because the first player can always win simply by
removing the sole counter.
Diagram 1-32 A Nim game in outcome class O
The Nim game in Diagram 1-32 is O because the second player can always win by
removing the remaining heap after the first player removes one of the heaps.
21
• •• •
Diagram 1-33 A Nim game in outcome class P
The Nim game in Diagram 1-33 is P because the third player can always win by
removing the last heap after the first and second players remove a heap in their respective
turns.
Propp established rules for classifying a three-player impartial game using a game tree.
• A game is N if at least one of its options is P.
• A game is O if all its options are N and has at least one option.
• A game is P if all its moves are O games.
• A game is if none of the above applies.
These outcome classes can also define three-player partizan games, but more outcome
classes are needed because left, middle, and right players have different moves in
partizan games. Alessandro Cincotti defined 27 outcome classes for three-player partizan
games when those games are defined as numbers using inequalities he introduced in his
paper [Cin_05]. Matt Klassen, however, suggests that there are more basic outcome
classes for three-player partizan games, 4,096 to be exact [Kla_10], which we will
explain below.
There are four possible results for each of the six turn orders after playing through a
three-player game: either left player (L) wins, right player (R) wins, middle player (M)
wins, or no player (N) can force a win, a total of 46 = 4,096 outcome classes. In the chart
below, we will show eight of the 4,096 possible outcome classes for a three-player
partizan combinatorial game.
22
Sample of Outcome Classes for Three-Player Partizan Combinatorial Games
Turn Order
Outcome Classes
N O P L R M ? LMR L M R L R M N M LRM L R M L R M N N MLR M L R L R M N R MRL M R L L R M N M RLM R L M L R M N L RML R M L L R M N N
Diagram 1-34
In the Diagram 1-34 chart, we listed ‘L,’ ‘M,’ ‘R,’ or ‘N’ for each outcome class’ turn
order. Those represent which of the three players has a winning strategy for that turn
order (or in N’s case, none of the players have a winning strategy). Note that we
introduced the M outcome class, where middle player can force a win in each turn order.
The reader may notice that we listed the last outcome class as ‘?’ on the table. This is
because, unlike the other seven outcome classes in Diagram 1-34, that outcome class
does not have an obvious label. The same can be said for most of the other 4,088
outcome classes not defined in this table. It is important to note, however, that all 4,096
outcome classes are non-empty, meaning that there is always a specific game for each
outcome class. To assist in demonstrating this, we will introduce L0, M0, R0, and N0.
Diagram 1-35
L0, M0, and R0 are specific three-player games where L, M, and R have a winning
strategy respectively. N0 is defined as the following game.
23
Diagram 1-36
The N0 tree indicates that no one can force a win, regardless of whether the player goes
first, second, or third. In other words, L0�œ L, M0�œ M, R0 œ R, and N0 œ .
These four games can be used to define any of the 4,096 outcome classes we introduced
earlier and show that none of them are empty games. We will demonstrate an example of
this by using the four games to define the ‘?’ outcome class listed in the Diagram 1-34
table.
Diagram 1-37
If you replace L0, M0, R0, and N0 in Diagram 1-37 with the game trees we defined in
Diagrams 1-34 and 1-35, you have created a game tree that defines who wins at each turn
order of the ‘?’ outcome class. The reader should note that the order of L0, M0, R0, and
N0 in the second level of Diagram 1-37’s game tree is the same as the order of results
listed for each turn order of the ‘?’ outcome class in Diagram 1-34. This process can be
done for all 4,096 outcome classes, meaning that all of them are non-empty.
The large number of outcome classes makes it more difficult to solve a three-player game
via its game tree. This is because there are too many outcome classes to create a chart
similar to the one in Diagram 1-26 that calculates the outcome class of a tree’s position
based on its children. However, it is still possible to solve a three-player combinatorial
24
game by showing for each turn order a player’s winning strategy or that no player can
force a win.
In the following pages, we’ll explain the three-player combinatorial game, 3D
Domineering, and how to solve that particular game.
What is 3D Domineering?
3D Domineering is a three-player variant of the Domineering game. It is played on a
rectangular solid board similar to a Rubik’s cube whose edges are parallel to the x, y, and
z axes. The three players have pieces that occupy two adjacent cubes of the board and
take turns in a cyclical order placing them in the 3D board. One player is limited to
placing pieces along the x-axis of the board, another player is limited to placing pieces
along the y-axis of the board, and the remaining player is limited to placing pieces along
the z-axis of the board. If a player is unable to place any more pieces in the board at his
turn, then that player is eliminated from the game. The remaining two players will
continue to play until one is unable place any more pieces in the board, resulting in a loss
for that player.
25
Example of a 3x3x3 Domineering Game
L’s piece
R’s piece
M’s piece M
• M
M •
••
Slice 1 Slice 2 Slice 3
Diagram 1-38
Diagram 1-38 shows an example of a 3x3x3 Domineering game. We marked the three
players as L, R, and M. Let’s suppose that the turn order for this game is M-L-R. M
plays a piece that traverses Slices 1 & 2 on the board. Then L and R play their pieces on
Slice 2. M follows up by playing a piece that traverses Slices 2 & 3. All three players
continue to take turns until two of the players are no longer able to place any more pieces.
Solving a 3D Domineering Game
A 3D Domineering game is considered solved when, for each turn order, one can either
identify the winning strategy for a player or prove that no player has a winning strategy.
Alessandro Cincotti was able to solve many A x B x C 3D Domineering games where A +
B + C < 10 and A, B, C ≥ 2 using an exhaustive search algorithm [Cin_08, Cin_09]. His
results are shown in the following chart (the chart presumes L, R, and M play on A, B,
and C respectively).
M
M
26
Outcome Classes for Three-Player Domineering L Playing First R Playing First M Playing First
2x2x2 M L R
3x2x2 L
2x3x2 R
2x2x3 M
3x3x2 M
2x3x3 L
3x2x3 R
3x3x3
4x2x2
2x4x2
2x2x4
4x3x2
2x4x3
3x2x4
4x2x3
3x4x2
2x3x4
5x2x2
2x5x2
2x2x5
Diagram 1-39
As of this writing, no analysis of 3D Domineering games where A + B + C ≥ 10 and A, B,
C ≥ 2 has been published yet. This is probably because the number of branches in a 3D
Domineering game tree becomes exponentially larger and more computationally
expensive to traverse as the 3D game board’s size increases. However, we have found
strategies that can significantly reduce the number of branches to traverse in a 3D
Domineering game tree.
Specifically, we found strategies for two players that can prevent the third player from
winning, regardless of the turn order and the third player’s actions. Selecting specific
moves for the two players, instead of considering all their possible moves, allows us to
prune branches from the 3D Domineering game tree. This makes it faster to traverse the
game tree and find the outcome class for 3D Domineering games, especially for those
whose sum of its dimensions is greater than or equal to ten. In the next chapter, we will
demonstrate how these strategies prove that a 3x3x3 Domineering game is a game (no
27
player can force a win in all turn orders). In subsequent chapters, we will expand on
those strategies to show that a 4x4x3 game, a game that has not been solved before, is
also a game.
28
2.0 – Strategies to Solve a 3x3x3 Domineering Game
In this chapter, we will present the strategies that will prove that a 3x3x3 Domineering
game is a game, where no one can force a win regardless of turn order. Alessandro
Cincotti had already solved the 3x3x3 Domineering game by using an exhaustive search
algorithm [Cin_08], but we will still present our strategies because they will serve as the
foundation for our proof to solve a 4x4x3 game.
In this chapter, we are going to prove the following:
Proposition 2-1: In a 3x3x3 game of Domineering, it is impossible for the middle player
(M) to force a win if both left player (L) and right player (R) collude to stop M.
Since a 3x3x3 game is symmetrical on all sides, if we prove that M cannot force a win if
L and R collude to stop M, then it also means that L cannot force a win if R and M
collude to stop L, and R cannot force a win if L and M collude to stop R. Thus, proving
Proposition 2-1 will also prove that no player can force a win in a 3x3x3 Domineering
game.
2.1 – Strategy to Block M in a 3x3x3 Domineering Game
There exists a strategy that L & R can follow that will prevent M from winning regardless
of whether M plays first, second, or third. To be clear, a win for M in a three-player
Domineering game means that M still has at least one legal move left after both L and R
have no more legal moves. There are two stages to this blocking strategy, the first which
we will demonstrate using the following examples of a 3x3x3 game:
29
L’s piece
R’s piece
M’s piece M
• M
M •
••
Slice 1 Slice 2 Slice 3
Diagram 2-1 An example of a 3x3x3 Domineering game
Notice that when M places a piece, it traverses either Slices 1 and 2 or Slices 2 and 3.
Observation: If all the spaces in Slice 2 are occupied, then M can no longer make any
more moves.
Therefore, the first part of the blocking strategy that L and R should adopt to prevent M
from winning is as follows:
Strategy: L & R should place as many pieces as possible on Slice 2 first before placing
any pieces on Slices 1 or 3.
Since L and R are going to be working together to stop M, we should consider this
scenario as a special two-player game between M and the L & R alliance. This means
that if M is unable to play any legal moves before both L and R are eliminated, then the
M
M
30
alliance wins. Furthermore, the alliance has an advantage over M where the alliance can
continue to play even if one of its members is unable to place any more pieces.
For this special two-player game, M’s move is considered a half-turn, and the other half-
turn is considered complete when the alliance finishes their turn. This holds true whether
L & R are both playing, or if only one player in the alliance is left playing against M.
Thus, if both L & R are playing, their moves are considered ¼ turns. However, if only
one member in the alliance is still playing, then those moves are considered half-turns.
We will demonstrate by example the alliance’s strategy to cover as much of Slice 2 as
possible using the following L-R-M game.
Start of game: L to play first
Slice 1 Slice 2 Slice 3
Diagram 2-2 The start of a 3x3x3 L-R-M Domineering Game
At the start of the game, Slice 2 has nine available squares, as shown in Diagram 2-2.
L to play after 1 turn
• •••
•
M
Slice 1 Slice 2 Slice 3
Diagram 2-3 A 3x3x3 Domineering game after one turn.
The other M piece is on Slice 1 or Slice 3, but for this example, its location is not important.
On the first turn (Turn 1), L, R, and M will each place a piece on Slice 2. At the end of
Turn 1, there are four spaces left in Slice 2 as shown in Diagram 2-3.
31
L to play after 2 turns
•• •••
••M ••
M • ••
Slice 1 Slice 2 Slice 3
Diagram 2-4 A 3x3x3 Domineering game after two turns.
Two one-space M pieces and an R piece are on Slice 1 and/or Slice 3
After the second turn (Turn 2), either L or R (in this example R) will be unable to place a
piece on Slice 2, forcing that player to play on Slice 1 or 3. The other two players will
place their pieces on Slice 2, leaving just one space left in Slice 2. At the start of the third
turn (Turn 3), M has only one more move available.
At this point, the alliance can no longer move on Slice 2, but it can still block M with the
second part of its blocking strategy:
Strategy: When there are no more available moves in Slice 2 for the alliance, L and R
can cooperatively block M’s remaining moves by placing pieces above and below M’s
available spaces on Slice 2.
We’ll use one possible game from Diagram 2-4 to demonstrate this strategy.
L to play after 2 turns
M••
M
Slice 1 Slice 2 Slice 3
Diagram 2-5 One possible game from Diagram 2-4.
•• •••
••M ••
M • ••
••••
••
32
In Diagram 2-5, M has one space left in Slice 2 and the alliance have an opportunity to
block that last space with their next move.
After 2.5 turns, M is eliminated
M••
•••• M
•• •••
••M ••
M ••••
••••
••
•••
Slice 1 Slice 2 Slice 3
Diagram 2-6 The alliance cooperatively blocks M’s last remaining move (as indicated by the ‘X’).
Diagram 2-6 shows that the alliance’s next moves block M from playing on the last
available space on Slice 2 (indicated by the ‘X’). M has no more available moves, which
results in a loss for M.
We have shown in the previous example how the alliance’s blocking strategy can prevent
M from winning, but our previous example only covers one possible 3x3x3 Domineering
game. To prove Proposition 2.1, we will need to show that the alliance’s two-part
blocking strategy prevents M from winning in all possible 3x3x3 games.
In the following sections, we’ll show specific moves that the alliance can play in a 3x3x3
Domineering game to win against M. For our examples, we’ll show all possible moves
for M, unless they can be ignored because of symmetry or because they would lead to an
immediate loss for M.
We begin our analysis with 3x3x3 games where M goes last.
2.2 – Analysis of L-R-M and R-L-M Games
In this section, we continue the proof of Proposition 2-1 by showing that the alliance can
prevent M from winning in 3x3x3 Domineering games where M goes last. We will start
by proving the following for 3x3x3 games:
33
Proposition 2-2: In L-R-M and R-L-M turn order games, the alliance can limit M to at
most one space in Slice 2 after two complete turns.
Proof:
Let’s examine how this is possible.
M to play after 0.5 turns
•• •••
••• •
• • •
Slice 1 Slice 2 Slice 3
Diagram 2-7 The alliance’s first moves on Slice 2.
For L-R-M and R-L-M games, Diagram 2-7 demonstrates the first possible moves the
alliance can make. We will show in subsequent examples that these moves are a good
first move for the alliance because M will not be able to force a win.
There are five possible ways that M can react to the alliance’s first moves in Diagram 2-
7, as demonstrated in the following diagram.
34
L (or R) to play after 1 turn on each board
• •• •••
••M
• •• •••
•• M
• •• •••
••
M
Potential Slice 2 Potential Slice 2 Potential Slice 2
• •• •••
••
M
• •• •••
••
M
Potential Slice 2 Potential Slice 2
Diagram 2-8 Each shaded square represents a possible move M can make after Diagram 2-7.
After M makes his first turn, the reader can see that in each possible game of Diagram 2-
8, there are four available spaces left in Slice 2. M playing on the center square in his
first turn (the first game in Diagram 2-8) is a really bad move for M because L and R are
able to cover the remaining spaces of Slice 2 in their next turns, eliminating M after 1.5
turns.
In the other four games of Diagram 2-8, one can see that only one of the alliance
members can move in the remaining spaces of Slice 2 in his next turn. The other alliance
member is forced to play on either Slice 1 or Slice 3. After either L or R moves on Slice
2 in those games, there are two available spaces left for M. When M moves on one of
those spaces, there will be one space left on Slice 2 after two complete turns. •
This concludes the proof of Proposition 2.2.
Although all the alliance needs to do after two turns is cooperatively block M’s remaining
move to eliminate M, we can actually prove the following:
35
Proposition 2-3: After the alliance limits M to one space in Slice 2 after two complete
turns, the alliance has at least one more move after M plays his final move even if L and
R do not cooperatively block M’s last move.
Proof:
The following chart shows the number of available spaces in each slice at every turn of a
specific L-R-M and R-L-M game.
Number of free spaces in each turn of a specific L-R-M & R-L-M game
Turn Number
L-R-M Game: Who Plays Next
R-L-M Game: Who Plays Next
# of Free Spaces
in Slice A
# of Free Spaces
in Slice 2
# of Free Spaces
in Slice B
Total # of Free Spaces
0 L to play next R to play next 9 9 9 27 0.25 R to play next L to play next 9 7 9 25
0.5 M to play next M to play next 9 5 9 23 1 L to play next R to play next 8 4 9 21
1.25 R to play next L to play next 8 2 9 19 1.5 M to play next M to play next 8 2 7 17
2 L to play next R to play next 7 1 7 15
2.25 R to play next L to play next 5 1 7 13 2.5 M to play next M to play next 5 1 5 11
3 L to play next R to play next 4 0 5 9
Diagram 2-9
There are a couple of things to note about the chart. First, the first part of the alliance’s
blocking strategy (L and R covering as much of Slice 2 as possible) is reflected in the
moves made in the chart. Second, for the columns “# of Free Spaces in Slice A” and “#
of Free Spaces in Slice B”, they can either be A = 1 and B = 3 or A = 3 and B = 1. Third,
the “Total # of Free Spaces” column can refer to all possible L-R-M and R-L-M games,
even if the chart refers to one specific game for each turn order. For example, let’s say
that after 1.5 turns, M decides to play on Slices 2 and B instead of Slices 2 and A.
36
Turn Number
L-R-M Game: Who Plays Next
R-L-M Game: Who Plays Next
# of Free Spaces
in Slice A
# of Free Spaces
in Slice 2
# of Free Spaces
in Slice B
Total # of Free Spaces
2 L to play next R to play next 8 1 6 15
2.25 R to play next L to play next 6 1 6 13
2.5 M to play next M to play next 6 1 4 11
3 L to play next R to play next 6 0 3 9
Diagram 2-10
Although the number of free spaces in Slices A and B are different in Diagram 2-10 than
what’s listed in Diagram 2-9, notice that the Total # of Free spaces are the same in both
charts at each turn order. This is because every piece each player places takes up two
spaces on the board. Regardless of how L, R, and M move in L-R-M and R-L-M games,
the total number of free spaces at each turn order will always be the same. Thus, after the
end of the second turn in L-R-M and R-L-M games, there will always be nine available
spaces left (assuming that L & R do not cooperatively block M’s remaining move).
The best way to prove that M loses in a three-player game is to show that either L or R
has at least one more move after M plays his final piece. Even if L and R do not block
M’s last move during their second turn, there are still nine available spaces between
Slices 1 and 3 after M plays his last move. With nine available spaces, it is guaranteed
that either L or R will be able to move at least once after M’s final turn, which is a loss
for M. •
This concludes the proof of Proposition 2.3.
The proof of Proposition 2.3 shows that the alliance can prevent M from winning 3x3x3
games where M goes last. The next section discusses 3x3x3 games where M goes
second.
37
2.3 – Analysis of L-M-R and R-M-L Games
In this section, we continue the proof of Proposition 2-1 by showing that the
alliance can prevent M from winning in 3x3x3 Domineering games where M goes
second. We will start by proving the following for 3x3x3 games:
Proposition 2-4: In L-M-R and R-M-L turn order games, the alliance can limit M to at
most two spaces in Slice 2 after 1.75 turns.
Proof:
Diagrams 2-11a and 2-11b show examples of L and R’s first moves in an L-M-R and an
R-M-L game respectively.
M to play after 0.25 turns M to play after 0.25 turns
•• •
••• •
• • •
• •••
• • •
• • •
Slice 2 (L-M-R game) Slice 2 (R-M-L game)
Diagram 2-11a Diagram 2-11b L’s first move R’s first move
We will show in subsequent examples that these moves are a good first move for the
alliance because M will not be able to force a win. The following examples in this
section show an L-M-R game, but they can also correlate to an R-M-L game because of
rotational symmetry (the reader can confirm this by looking at Diagrams 2-11a and 2-
11b).
There are seven possible ways that M can react to L’s (or R’s) first move in an L-M-R (or
R-M-L) game, which are shaded in the seven games of the following diagram. We also
demonstrate how the alliance can react to each of M’s seven possible first moves.
38
M to play after 1.25 turns on each board (all represent Slice 2)
• M •• •
••
• • M
• •
••
•• M ••• •
• ••• • M
•
Game 1• Game 2• Game 3• Game 4•
• ••• •
M •
• ••• •
M •
• •• •
•• M
Game 5• Game 6• Game 7•
Diagram 2-12 M’s seven possible first moves and the alliance’s reaction to those moves.
Note that in each of the seven games in Diagram 2-12, there are two remaining spaces
left in Slice 2. After M moves on one of those spaces in his next turn, there will be just
one space left after 1.75 turns. •
This concludes the proof of Proposition 2-4.
Although all the alliance needs to do after 1.75 turns is cooperatively block M’s
remaining move to eliminate M, we can actually prove the following:
Proposition 2-5: After the alliance limits M to one space in Slice 2 after 1.75 turns, the
alliance has at least one more move after M plays his final move even if L and R do not
cooperatively block M’s last move.
Proof:
To prove this proposition, we will use a chart similar to the one in Diagram 2-9 to show
the number of available spaces left in each slice after each turn in a specific L-M-R and
R-M-L game.
39
Number of free spaces in each turn of a specific L-M-R & R-M-L game
Turn Number
L-M-R Game: Who Plays Next
R-M-L Game: Who Plays Next
# of Free Spaces
in Slice A
# of Free Spaces
in Slice 2
# of Free Spaces
in Slice B
Total # of Free Spaces
0 L to play next R to play next 9 9 9 27 0.25 M to play next M to play next 9 7 9 25
0.75 R to play next L to play next 8 6 9 23
1 L to play next R to play next 8 4 9 21 1.25 M to play next M to play next 8 2 9 19
1.75 R to play next L to play next 7 1 9 17 2 L to play next R to play next 7 1 7 15
2.25 M to play next M to play next 5 1 7 13 2.75 R to play next L to play next 4 0 7 11
Diagram 2-13
There are a couple of things to note about the chart. First, the first part of the alliance’s
blocking strategy (L and R covering as much of Slice 2 as possible) is reflected in the
moves made in the chart. Second, for the columns “# of Free Spaces in Slice A” and “#
of Free Spaces in Slice B”, they can either be A = 1 and B = 3 or A = 3 and B = 1. Third,
the “Total # of Free Spaces” column can refer to all possible L-M-R and R-M-L games
even if the chart refers to one specific game for each turn order. Recall that we
demonstrated in the Diagram 2-9 and 2-10 charts that, regardless of where L, R, and M
play, each player’s piece takes up two spaces.
To reiterate what was stated in the last section, the best way to prove that M loses in a
three-player game is to show that either L or R has at least one more move after M plays
his final piece. Even if L and R do not block M’s last move after 1.75 turns, there are
still eleven available spaces between Slices 1 and 3 after M plays his last move. With
eleven available spaces, it is guaranteed that either L or R will be able to move at least
once after M’s final turn, which is a loss for M. •
This concludes the proof of Proposition 2.5.
The proof of Proposition 2.5 shows that the alliance can prevent M from winning 3x3x3
games where M goes second. The next section discusses 3x3x3 games where M goes
first.
40
2.4 – Analysis of M-L-R and M-R-L Games
We have proven in the last two sections that the alliance can prevent M from winning
3x3x3 Domineering games whenever M goes second or third. If we prove that the same
applies to games where M goes first, then we have demonstrated that the alliance can
prevent M from winning all possible 3x3x3 games. In this section, we conclude the proof
of Proposition 2-1 by proving the following for 3x3x3 games:
Proposition 2-6: In M-L-R and M-R-L turn-order games, the alliance can limit M to a
maximum of three spaces in Slice 2 after two turns.
Proposition 2-7: In M-L-R and M-R-L turn-order games, the alliance has at least one
more move after M plays his final piece.
There are nine starting moves for M in 3x3x3 games where M goes first. However, we
can break those down to three types of moves, as demonstrated in Diagram 2-14.
L (or R) to play after 0.5 turns on each board
Game 1 Game 2 Game 3
M
M
M
Slice 2 Slice 2 Slice 2
Diagram 2-14 The three types of moves M can play when he goes first.
The first type of move M can make is playing on a corner space. M playing on any of the
shaded spaces in Game 1 of Diagram 2-14 represents the same move because of
rotational symmetry. The second type of move M can make is playing on the center
space, as shown in Game 2 of Diagram 2-14. The third type of move M can make is
playing on a space that is neither corner nor center. M playing on any of the shaded
squares in Game 3 of Diagram 2-14 represents the same move because of rotational
41
symmetry. Therefore, we only need to show these three games in our subsequent
examples.
We will continue the proof of Proposition 2-6 with demonstrating the first two types of
moves M can make.
L (or R) to play after 0.5 turns L (or R) to play after 0.5 turns
M
•
M
M M
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-15a Diagram 2-15b M plays on the upper-left corner. M plays on the center square.
In Diagram 2-15a, M plays on a corner space. In Diagram 2-15b, M plays on the center
space. In both games, M’s piece traverses Slices 1 and 2, but M could also place a piece
that traverses Slices 2 and 3. However, we do not need to show the games where M’s
piece traverses Slices 2 and 3 because they are symmetrical to the games where M’s
piece traverses Slices 1 and 2.
The following shows how the alliance can respond to M’s corner and center moves.
M to play after 1 turn M to play after 1 turn
M
••• •
M ••• •
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-16a Diagram 2-16b The alliance responds to M’s corner move. The alliance responds to M’s center move.
We will show in subsequent examples that these moves are a good first move for the
alliance because M will not be able to force a win. In both diagrams, there are four
available spaces left on Slice 2 after one complete turn.
M
• M
42
M to play after 1 turn M to play after 1 turn
Slice 2 Slice 2
Diagram 2-17a Diagram 2-17b The remaining playable spaces on The remaining playable spaces on
the Slice 2 from Diagram 2-16a. the Slice 2 from Diagram 2-16b.
In each game, M has four possible moves in the remaining spaces of Slice 2, which are
indicated in the following diagrams.
L (or R) to play after 1.5 turns
M M
Slice 2 Slice 2 Slice 2 Slice 2
Diagram 2-18 This shows all four possible places where M can place a piece on Slice 2 from Diagram 2-17a.
The other M piece is on either Slice 1 or Slice 3.
L (or R) to play after 1.5 turns
M
M M M
Slice 2 Slice 2 Slice 2 Slice 2
Diagram 2-19 This shows all four possible places where M can place a piece on Slice 2 from Diagram 2-17b.
The other M piece is on either Slice 1 or Slice 3.
One can see in Diagrams 2-18 and 2-19 that regardless of where M places his next piece,
only one alliance member can place a piece on Slice 2 in his next turn. The other must
place his piece on either Slice 1 or 3. This means that three more spaces on Slice 2 will
be occupied at the end of the second turn, leaving M with only one more space to play.
M M
43
This partially proves Proposition 2-6, and we will complete this proof later in this section.
Until then, we will prove that M will not be able to force a win after playing on a corner
or center space using a chart similar to the ones in Diagrams 2-9 and 2-13. This chart
shows the number of available spaces left in each slice after each turn in an M-L-R and
M-R-L game.
Number of free spaces in each turn of a specific M-L-R & M-R-L game
Turn Number
M-L-R Game: Who Plays Next
M-R-L Game: Who Plays Next
# of Free Spaces
in Slice A
# of Free Spaces
in Slice 2
# of Free Spaces
in Slice B
Total # of Free Spaces
0 M to play next M to play next 9 9 9 27
0.5 L to play next R to play next 8 8 9 25
0.75 R to play next L to play next 8 6 9 23
1 M to play next M to play next 8 4 9 21
1.5 L to play next R to play next 7 3 9 19 1.75 R to play next L to play next 7 1 9 17
2 M to play next M to play next 7 1 7 15
2.5 L to play next R to play next 6 0 7 13
Diagram 2-20
There are a couple of things to note about the chart. First, the first part of the alliance’s
blocking strategy (L and R covering as much of Slice 2 as possible) is reflected in the
moves made in the chart. Second, for the columns “# of Free Spaces in Slice A” and “#
of Free Spaces in Slice B”, they can either be A = 1 and B = 3 or A = 3 and B = 1. Third,
the “Total # of Free Spaces” column can refer to all possible M-L-R and M-R-L games
even if the chart refers to one specific game for each turn order. Recall that we
demonstrated in the Diagram 2-9 and 2-10 charts that, regardless of where L, R, and M
play, each player’s piece takes up two spaces.
Diagram 2-20 shows games where the second player is able to move on Slice 2 during the
second turn. The following chart shows what happens if the third player is able to move
on Slice 2 during the second turn instead.
44
Number of free spaces in each turn of a specific M-L-R & M-R-L game
Turn Number
M-L-R Game: Who Plays Next
M-R-L Game: Who Plays Next
# of Free Spaces
in Slice A
# of Free Spaces
in Slice 2
# of Free Spaces
in Slice B
Total # of Free Spaces
0 M to play next M to play next 9 9 9 27 0.5 L to play next R to play next 8 8 9 25
0.75 R to play next L to play next 8 6 9 23 1 M to play next M to play next 8 4 9 21
1.5 L to play next R to play next 7 3 9 19 1.75 R to play next L to play next 7 3 7 17
2 M to play next M to play next 7 1 7 15
2.5 L to play next R to play next 6 0 7 13
Diagram 2-21
Although the number of free spaces in Slices A and B deviates in Turns 1.5 and 1.75
from those in Diagram 2-21, the total number of free spaces for each turn in both charts
remains the same.
To reiterate what was stated in the last two sections, the best way to prove that M loses in
a three-player game is to show that either L or R has at least one more move after M
plays his final piece. After two complete turns, there is one space left in Slice 2. After M
moves on that space in his last turn, there are thirteen available spaces left in Slices A and
B. With thirteen available spaces, it is guaranteed that either L or R will be able to move
at least once after M’s final turn, which is a loss for M.
This analysis covers all M-L-R and M-R-L games where M plays on either the corner or
center square first, so we have partially proved Proposition 2-7. Next, we will analyze all
M-L-R and M-R-L games where M’s first move is neither corner nor center (the third
type of move in Diagram 2-14) to complete the proofs of Propositions 2-6 and 2-7.
45
L (or R) to play after 0.5 turns
M
Slice 1 Slice 2 Slice 3
Diagram 2-22 M’s first move is on a space that is neither corner nor center.
Diagram 2-23 demonstrates where the alliance can play after M’s move on Diagram 2-22.
We will show in subsequent examples that these moves are a good first move for the
alliance because M will not be able to force a win.
M to play after 1 turn
M M
••• •
Slice 1 Slice 2 Slice 3
Diagram 2-23 The alliance responds to M’s move from Diagram 2-22
There are four spaces left in Diagram 2-23. M has four possible moves in the remaining
spaces of Slice 2, which are shaded in the following diagram.
L (or R) to play after 1.5 turns
M
M M
M
Slice 2 Slice 2 Slice 2 Slice 2
Diagram 2-24 This shows all four possible places where M can place a piece on the Slice 2 from Diagram 2-23.
The other M piece is on either Slice 1 or Slice 3.
46
One can see that the second possible game marked in Diagram 2-24 prevents the alliance
from playing any more pieces on Slice 2. The other three possibilities allow one alliance
member to move in Slice 2 in their next turn, leaving M with one move left after two
turns. This creates a losing scenario for M similar to the games where M played on a
corner or center square (see the charts in Diagrams 2-20 and 2-21). The move in
Diagram 2-24’s second game is a better move for M than the other games because it
leaves M with three available spaces on Slice 2 after two turns instead of one available
space. Therefore, we will continue our analysis from that second possible game.
This also concludes the proof of Proposition 2.6.
In our previous examples, we were able to show that the alliance could prevent M from
winning without having to do the second part of their blocking strategy (playing pieces
above and below available M spaces). However, now that M can have three available
spaces in Slice 2 after 1.5 turns, the alliance will have to execute the second part of their
blocking strategy to prevent M from winning. We will demonstrate this in subsequent
examples.
L (or R) to play after 1.5 turns L (or R) to play after 1.5 turns
M
•
M
M ••• •
M
M
M •
M
M ••• •
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-25a Diagram 2-25b M plays second piece on Slices 2 and 3 from M plays second piece on Slices 1 and 2 from
Diagram 2-24’s second possible game. Diagram 2-24’s second possible game.
From Diagram 2-24’s second game, there are two possible ways M can place his second
piece—the piece can either traverse Slices 2 and 3 (see Diagram 2-25a) or it can traverse
Slices 1 and 2 (see Diagram 2-25b).
Note: If we can show that the alliance can prevent M from winning in these two cases,
we will have finished considering all possible M moves and prove Proposition 2.7.
Let’s examine how each of these games continue, starting with Diagram 2-25a.
47
••
L (or R) to play after 1.5 turns
M M
M ••• •
M
Slice 1 Slice 2 Slice 3
Diagram 2-26 From Diagram 2-25a, M played his second piece across Slices 2 & 3
M has three available spaces to play on Slice 2 after playing his second piece. However,
the alliance can perform the second half of their blocking strategy and block one of those
spaces by playing on squares above and below an available space.
M to play after 2 turns
••M ••M M ••••••••
••••
M
Slice 1 Slice 2 Slice 3
Diagram 2-27 The ‘X’ indicates that M can no longer play on that square because L and R blocked M by playing pieces above and below the square.
The alliance has blocked M from playing on the upper-left corner of Slice 2 by playing
their pieces on the upper-left corner of Slices 1 & 3 during their turn. This now leaves M
with two available spaces in Slice 2 instead of three. M has four possible moves left, but
they can be reduced to one.
48
L (or R) to play after 2.5 turns L (or R) to play after 2.5 turns
M
M
Slice 1 Slice 3 Slice 1 Slice 3
Diagram 2-28a Diagram 2-28b From Diagram 2-27, M plays on the center square. From Diagram 2-27, M plays on the upper-right square.
It also shows the remaining spaces for L & R. It also shows the remaining spaces for L & R.
Diagrams 2-28a and 2-28b show two possible games where M plays on one of the
available squares on Slice 2. In both diagrams, M’s piece traverses Slices 1 and 2. Note
that the remaining available spaces for M to play in Diagram 2-27 are the same in Slices
1 and 3. Thus, M playing on Slices 2 and 3 is the same as playing on Slices 1 and 2,
which is why we only show two examples in Diagram 2-28 instead of four.
In Diagram 2-28a, one can see that after M plays on the center square, M still has one
move left. This is because the alliance is unable to block the upper-right corner of Slice
2. It is impossible for R to play on the upper-right corner of Slice 1 or 3. On the other
hand, if M plays on the upper-right corner as shown in Diagram 2-28b, then the alliance
can block M’s final move in Slice 2’s center square by placing their pieces over the
center square of Slices 1 and 3, thus eliminating M from the game. Therefore, the better
move for M is to play on the center square.
L (or R) to play after 2.5 turns
••M ••M
M
M M ••• •
••••
M
Slice 1 Slice 2 Slice 3
Diagram 2-29 From Diagram 2-28a,
M has played his second piece across Slices 1 & 2 on the center square
49
Since M playing on Slices 2 and 3 is symmetrical to M playing on Slices 1 and 2, there is
no need to show the game where M plays on Slices 2 and 3 in our subsequent examples.
M now has only one space left to play on Slice 2.
M to play after 3 turns
••M ••••M ••
M
M M ••• •
•••
M
•••Slice 1 Slice 2 Slice 3
Diagram 2-30 The alliance responds to M’s move in Diagram 2-29.
The alliance is unable to block M from playing on the upper-right corner, but L will force
M to play on only Slices 2 and 3 by placing his piece over the upper-right corner of Slice
1.
L (or R) to play after 3.5 turns
••M ••••M ••
M M
M M ••• •
•••M
M
•••Slice 3 Slice 1 Slice 2
Diagram 2-31 M plays his final piece.
One can see that after M plays his final piece on the upper-right corner, both L and R can
still play on the remaining available spaces. This is a loss for M because the alliance has
at least one legal move left in the combined six available spaces of both Slices 1 and 3
after M plays his final piece. The previous examples apply to both M-L-R and M-R-L
turn-order games. We have covered the game from Diagram 2-25a, so let’s examine the
game from Diagram 2-25b.
50
•••
L (or R) to play after 1.5 turns
M
M
M
M ••• •
Slice 1 Slice 2 Slice 3
Diagram 2-32 From Diagram 2-25b, M has played his second piece across Slices 1 & 2.
M has three available spaces to play in Slice 2, so the alliance will perform the second
part of their blocking strategy by playing on space above and below an available M
space. However, M has placed his two pieces in such a way that L and R cannot to block
the upper-left corner of Slice 2 at any point during the game. This tactic is called
reserving a space, and places M in a better position than the game in Diagram 2-25a
because M has guaranteed that one of his moves will not be blocked. However, the
alliance is still capable of blocking M’s other spaces.
M to play after 2 turns
M •
M •M•
M ••• •
Slice 1 Slice 2 Slice 3
Diagram 2-33 The ‘X’ indicates that M can no longer play on that square because L and R blocked M by playing pieces above and below the square.
When L and R covered the upper-right corner of Slices 1 and 3, it removes the upper-
right corner of Slice 2 as a playable space for M. This leaves M with two more available
moves on Slice 2. If M were to play on the reserved upper-left corner of Slice 2, then the
alliance can cooperatively block the spaces above and below M’s last available move in
Slices 2’s center space, which would result in a loss for M. However, the alliance cannot
block the upper-left corner of Slice 2 because it is reserved for M. Therefore, it is a better
move for M to play on the center square because M will still have one more move after
he plays on the center square.
51
••
••
••
••
••
L (or R) to play after 2.5 turns L (or R) to play after 2.5 turns
M •
M M •M
M M ••• •
M •
M •M
M M ••• •
••
M
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-34a Diagram 2-34b M plays on the unprotected center square. M plays on the unprotected center square.
Diagram 2-34a shows M playing a piece on the center square that traverses Slices 1 and
2. Diagram 2-34b shows M playing a piece on the center square that traverses Slices 2
and 3. Our following examples will follow these two games.
M to play after 3 turns M to play after 3 turns
M •
M M •M
M M ••• •
••
M •
M •M
M M ••• •
• ••• M
••
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-35a Diagram 2-35b The alliance responds to The alliance responds to
M’s move in Diagram 2-34a. M’s move in Diagram 2-34b.
After M’s third turn, L and R place their pieces on Slice 3. In both Diagrams 2-35a and
2-35b, L covering the upper-left corner of Slice 3 forces M to play his last piece on the
upper-left corner of Slices 1 & 2.
L (or R) to play after 3.5 turns L (or R) to play after 3.5 turns
M M •
M M••M M
M M ••• •
••
M M •
M •M M
M M ••• •
• ••• M
••
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 2-36a Diagram 2-36b: M played his final move. M played his final move;
In both Diagrams 2-36a and 2-36b, one can see that after M plays his final piece, there
are still six available spaces that the alliance can play on. Since the alliance is still able to
move after M plays his final piece, then M has lost the game.
52
This concludes our analysis of M-L-R and M-R-L games where M plays on a space that
is neither corner nor center first. •
•
This also concludes the proof of Proposition 2-7.
This completes our coverage of all possible 3x3x3 Domineering games. We have proved
that regardless of whether M plays first, second, or third, M can never force a win in a
3x3x3 Domineering game if L and R collude to stop him using the strategies we
described in this chapter.
This completes the proof of Proposition 2-1.
In other words, a 3x3x3 Domineering game is in the outcome class .
53
3.0 – Strategies to Solve a 4x4x3 Domineering Game: Part 1
In Chapters 3 and 4, we are going to extend the strategies we defined in the previous
chapter to solve a 4x4x3 Domineering game. Since a 4x4x3 game is not symmetric on all
sides, this affects our analysis of where L, R, and M play in that game’s board. We
mentioned in the first chapter that the edges of a 3D Domineering board are parallel to
the x-, y-, and z-axes. For this thesis, we will define where L, R, and M play along those
axes.
Diagram 3-1 A 4x4x3 board aligned with the x, y, and z axes.
R’s move (highlighted in dark gray) is parallel to the x-axis.
In Diagram 3-1, we establish that R’s moves will be parallel to the x-axis (example
highlighted in dark gray). In this thesis, we consider the A component of an A x B x C
game to be parallel to the x-axis. Therefore, R will be playing on the first 4 component
of a 4x4x3 game according to our definition, regardless of turn-order.
54
Diagram 3-2 A 4x4x3 board aligned with the x, y, and z axes.
L’s move (highlighted in dark gray) is parallel to the y-axis.
In Diagram 3-2, we establish that L’s moves will be parallel to the y-axis (example
highlighted in dark gray). In this thesis, we consider the B component of an A x B x C
game to be parallel to the y-axis. Therefore, L will be playing on the second 4
component of a 4x4x3 game according to our definition, regardless of turn-order.
Diagram 3-3 A 4x4x3 board aligned with the x, y, and z axes.
M’s move (highlighted in dark gray) is parallel to the z-axis.
In Diagram 3-3, we establish that M’s moves will be parallel to the z-axis (example
highlighted in dark gray). In this thesis, we consider the C component of an A x B x C
game to be parallel to the z-axis. Therefore, M will be playing on the 3 component of a
4x4x3 game according to our definition, regardless of turn-order.
We established these definitions to make it clear where M is playing when presenting the
proof for solving a 4x4x3 game. In Chapter 3, we will show that the alliance can prevent
M from winning a 4x4x3 Domineering game if M plays on the 3 component. In Chapter
55
4, we will show that the alliance can prevent M from winning a 4x4x3 Domineering
game if M plays on the second 4 component (which will be subsequently be referred to as
a 4x3x4 game in this thesis).
In this chapter, we are going to prove the following:
Proposition 3-1: M is unable to force a win in a 4x4x3 game, where M plays on the 3
component, if the L & R alliance teams up against him.
The strategies that L & R will use to do this are similar to the strategies that prevented M
from winning in a 3x3x3 board. Those strategies are to cover up as much of Slice 2 as
possible to limit the number of available moves M has on the board. Then the alliance
covers the spaces above and below any remaining spaces in Slice 2 to eliminate M’s
remaining moves.
To start the proof of Proposition 3.1, we will prove the following for 4x4x3 Domineering
games where M plays on the 3 component:
Proposition 3-2: Regardless of turn order, when the alliance covers as much of Slice 2
as possible, the maximum number of available spaces M can play on Slice 2 is three,
assuming best play from M and the alliance.
3.1 – Analysis of L-R-M and R-L-M Games
To prove Proposition 3-2, we will prove the following for 4x4x3 Domineering games
where M plays on the 3 component:
Proposition 3-3: In L-R-M and R-L-M turn order games, the alliance can limit the
number of M’s available spaces in Slice 2 to a maximum of two after 2.5 turns.
Let’s examine how the alliance can accomplish that.
56
M to play after 0.5 turns
• ••••
Slice 1 Slice 2 Slice 3
Diagram 3-4 The alliance plays on the upper-left corner of Slice 2.
In Diagram 3-4, the alliance played on the upper-left corner of Slice 2. By the end of this
chapter, we will show that these are a good first move for the alliance because M will not
be able to force a win. M’s next move will cover one of the twelve remaining spaces in
Slice 2, so our next examples will cover all of those twelve possibilities, starting with M
playing on Row 2, Column 4 in Diagram 3-2.
Diagram 3-5 shows M’s piece traversing Slices 1 and 2, but in this and subsequent
sections (3.2 and 3.3), it does not matter if M’s piece traverses Slices 1 and 2 or Slices 2
and 3. We are more concerned with how Slice 2 is populated in the first turns of a 4x4x3
game. Nonetheless, we will address how the endgame of a 4x4x3 game is affected by
whether a piece is placed on Slices 1 and 2 or Slices 2 and 3 in Section 3.4.
L (or R) to play after 1 turn
M
• •••• M
Slice 1 Slice 2 Slice 3
Diagram 3-5 M’s move on Slice 2 prevents L & R from playing on the upper-right corner of the board.
57
•••• ••••
This appears to be a good move for M because he reserved a space (meaning that the
alliance cannot place a piece in Slice 2 to cover it). However, this situation is okay for
the alliance because it will limit M to reserve only one more space in Slice 2.
M to play after 1.5 turns
M
•••••••• M
••••••
Slice 1 Slice 2 Slice 3
Diagram 3-6 From Diagram 3-5, L & R play pieces that create a 2x3 rectangular area (shaded above)
In Diagram 3-6, L & R have placed pieces that create a 2x3 rectangular area in Slice 2,
which is shaded in the above diagram. This area allows both alliance members to play on
Slice 2 in their next turn regardless of where M plays next.
M to play after 2.5 turns on each board
M ••• ••
Potential area of Slice 2 Potential area of Slice 2 Potential area of Slice 2
•• •
M ••• ••••M
• •••• M
Potential area of Slice 2 Potential area of Slice 2 Potential area of Slice 2
Diagram 3-7 Six possible outcomes for the shaded squares in Diagram 3-6.
Note that in each of the above cases, M is left with just one available space in the 2x3
rectangular area. That one available space and the reserved space in the upper-right
corner of Diagram 3-6 show that the alliance is able to restrict M to two available spaces
on Slice 2 after 2.5 turns. However, this example only applies to M playing on Row 2,
M• M•
58
Column 4 as his first move. We’ll next review M playing on Row 2, Column 2 and Row
1, Column 4 in the following diagrams.
L (or R) to play after 1 turn L (or R) to play after 1 turn
M
• ••• M •
•
• • •
M • •• M
• •
•
• • •
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 3-8a Diagram 3-8b After Diagram 3-4, After Diagram 3-4,
M plays on Row 2, Column 2 M plays on the upper-right corner
The next diagrams show how the alliance can respond to the moves M made in Diagrams
3-8a and 3-8b.
M to play after 1.5 turns M to play after 1.5 turns
M
• ••• M ••
• •
• • • •
M • •• M
• ••
• •
• • • •
Slice 1 Slice 2 Slice 3 Slice 1 Slice 2 Slice 3
Diagram 3-9a Diagram 3-9b After Diagram 3-8a, After Diagram 3-8b,
L & R play pieces to create a L & R play pieces to create a 2x3 rectangular area (shaded above) 2x3 rectangular area + 1 (shaded above)
In Diagram 3-9a, although the alliance helped M reserve a spot in the upper-right corner,
their moves created a 2x3 rectangular area (shaded in Diagram 3-9a). We have already
demonstrated in Diagram 3-7 that this area will permit M to reserve only one more space,
as both alliance members can move in Slice 2 in their next turn. This leaves M with a
total of two available spaces in Slice 2 after 2.5 turns.
In Diagram 3-9b, the alliance created a 2x3 rectangular area with an additional space
appended to it (shaded in Diagram 3-9b). If we treat this area like the 2x3 rectangular
area in Diagram 3-9a and presume that the appended space is a reserved space for M,
then we will arrive at the same conclusion as Diagram 3-9a—the alliance has left M with
only two available spaces in Slice 2 after 2.5 turns.
59
The next diagrams demonstrate what happens if M plays on Row 2, Column 3 as his first
move. This covers the fourth possible staring move for M after Diagram 3-4.
M to play after 1.5 turns M to play after 2.5 turns on each board
M
• •• •• M •
••
• • •
M
•• ••
• M
• •• •M •
• ••
Slice 1 Slice 2 Slice 3
Diagram 3-10a After Diagram 3-4,
M plays on Col 3, Row 2 L & R respond and created
seven contiguous spaces (shaded above)
••
M •• •
••
M•••
••
•• M• •
••
•• M•
Diagram 3-10b Regardless of where M plays after
Diagram 3-10a, there will always be left two spaces left in Slice 2.
If M plays on Column 3, Row 2 as shown in Diagram 3-10a, the alliance can create an
area of seven contiguous spaces in their next move (shaded in Diagram 3-10a). As
indicated in Diagram 3-10b, the alliance members are both able to move in their next turn
regardless of where M plays on that shaded area, leaving M with two available spaces in
Slice 2 after 2.5 turns.
So far, we have demonstrated in Diagrams 3-6 to 3-10b that the alliance can hold M to
two available spaces in Slice 2 after 2.5 turns if M were to play on any of the top-half
squares of Slice 2 in his first turn.
This leaves eight more possible moves for M to play in his first turn. Those remaining
moves are on the lower half of Slice 2, and regardless of where M plays on that area, the
alliance will always be able to create a 2x4 rectangular area minus one square by playing
on the top half of Slices 2, as shown in the following diagram.
60
M to play after 1.5 turns on each board
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
• •• •• •• •
M
Diagram 3-11 Demonstrates eight possible moves for M on Slice 2’s lower half.
L & R playing on top half creates 2x4 area minus one area (shaded).
The reader can verify in Diagram 3-11 that regardless of where M plays next in the
shaded area of all eight possible games, the alliance members will both be able to play on
Slice 2 in their next turns. This leaves M with two available spaces in Slice 2 after 2.5
turns.
We have exhausted all possible moves for M to play on his first turn after the alliance’s
first moves. We have shown that for the twelve possible first-turn M moves on Slice 2,
the alliance will leave M with at most two available spaces in Slice 2 in L-R-M and R-L-
M turn-order games after 2.5 turns.
This concludes the proof of Proposition 3-3.
3.2 – Analysis of M-L-R and M-R-L Games
In this section, we will continue the proofs of Propositions 3-1 and 3-2 by proving the
following for 4x4x3 Domineering games where M plays on the 3 component:
Proposition 3-4: In M-L-R and M-R-L turn order games, the alliance can limit the
number of M’s available spaces in Slice 2 to a maximum of three after 3 turns.
61
There are sixteen starting moves for M in an M-L-R and M-R-L game. However, we can
break this down to three types of moves thanks to symmetry, as demonstrated in Diagram
3-12.
L (or R) to play after 0.5 turns on each board
Game 1 Game 2 Game 3
M
• M
•
M
Slice 2 Slice 2 Slice 2
Diagram 3-12 Taking symmetry into account, M has three starting moves in a M-L-R (or M-R-L) game.
Symmetric squares are shaded.
The first board of Diagram 3-12 (Game 1) represents M playing on any of the corner
spaces (which are shaded). The second board (Game 2) represents M playing on one of
the four center squares of Slice 2 (which are shaded). The third board (Game 3)
represents M playing on spaces that are neither corner nor center (which are shaded).
In order for the alliance to restrict M to a maximum of three reserved spaces after two
turns, they have to be careful not to allow M to reserve two spaces before M’s second
turn and three spaces before M’s third turn.
62
M to play after 1 turn on each board Game 1 Game 2 Game 3
• •• M
••• ••••M
•
• M ••••
A possible Slice 2 A possible Slice 2 A possible Slice 2
Diagram 3-13 From Diagram 3-12, these diagrams show where the alliance should play after M’s first move.
Diagram 3-13 shows how the alliance should respond to the three types of starting moves
M can make. By the end of this chapter, we will show that these are a good first move
for the alliance because M will not be able to force a win. In Games 1 and 3, the reader
can see that the alliance has not left M any openings to reserve a space in his second turn.
However, M can reserve a space in Game 2 by playing on Row 2, Column 4 in his second
turn. M playing on one of the center four squares is actually M’s best move, because
regardless of where the alliance places their pieces in their next turns, M will always be
able to reserve a space in his second turn. However, because we will prove that the
alliance’s initial move in Diagram 3-13’s Game 2 is a win for the alliance, the analysis of
all other options for L & R in Game 2 is not necessary.
For the purpose of proving that the alliance can hold M to a maximum of three reserved
spaces, let’s examine all three possible starting moves for M, beginning with M playing
on a corner space (Game 1).
63
• • • •
M to play after 2 turns on each board (all represent Slice 2)
• •• M
• •• M
••
• •• M
• ••
• M
•
• •• M
• •• ••
M
• •• M
• M ••
••
• •• M
• ••
• M
•
• •• M
• ••
•• M
• •• M
• M ••
••
• •• M
• •• •
M •
• •• M
• ••
•
M •
• •• M
• •
M •
••
• •• M
•
•
M •• •
Diagram 3-14 These boards showcase the eleven possible moves M can play after Diagram 3-13’s Game 1
(which are shaded). Also shown is the alliance’s response to each shaded move.
Diagram 3-14 shows all eleven possible moves that M can make from Diagram 3-13’s
Game 1 configuration. Also shown is how the alliance should respond to those shaded
moves in each board.
Note that after two turns, there are six spaces left in every board and no reserved moves
yet for M. We will examine those spaces after we cover the possible board
configurations for Diagram 3-13’s Games 2 and 3.
Now let’s examine the possible moves for M when he plays on one of the center squares
(Diagram 3-13’s Game 2):
64
• • • •
M to play after 2 turns on each board (all represent Slice 2)
• •• M
• M ••
••
• ••
• M • M
•• •
• •• •• M •
M
••
• ••
• M
•• •• M
M’s best move
• •• •• M M •
••
• ••
• M
• M
• ••
• ••
• M
•
•• M •
• ••
• M
• M
• ••
• •• •• M •
M ••
• ••
• M
M •
•• •
• •• •• M •
M ••
Diagram 3-15 These boards showcase the eleven possible moves M can play after Diagram 3-13’s Game 2
(which are shaded). Also shown is the alliance’s response to each shaded move.
Diagram 3-15 shows all eleven possible moves that M can make from Diagram 3-13’s
Game 2. Also shown is how the alliance should respond to those shaded moves in each
board.
Note that after two turns, there are six spaces left in every board. The board marked
“M’s best move” is indeed M’s best move because it is the only one that reserves a space
for M, leaving the alliance at a disadvantage at the start of Turn 2. We’ll analyze the
remaining spaces from Diagrams 3-14 and 3-15 after examining the possible moves from
Diagram 3-13’s Game 3 scenario.
65
• • • •
M to play after 2 turns on each board (all represent Slice 2)
• M ••
• •• M
••
• M ••
• ••
• M
•
• M ••
• •• ••
M
• M ••
• M ••
••
• M ••
• •• •
M •
• M ••
• ••
•
M •
• M ••
• M ••
••
• M ••
• •• •
M •
• M ••
• ••
•• M
• M ••
• •
M •
••
• M ••
• •• ••
M
Diagram 3-16 These boards showcase the eleven possible moves M can play after Diagram 3-13’s Game 3
(which are shaded). Also shown is the alliance’s response to each shaded move.
Diagram 3-16 shows all eleven possible moves that M can make from Diagram 3-13’s
Game 3 configuration. Also shown is how the alliance should respond to those shaded
moves in each board.
Note that after two turns, there are six spaces left in every board. Some of the remaining
spaces in this diagram are similar to remaining spaces in Diagrams 3-14 and 3-15.
Removing symmetrical cases, the following is a collection of remaining spaces from
Diagrams 3-14 to 3-16.
66
• • •
M to play after 2 turns on each board
•
M’s best move•Diagram 3-17
The above are all six-space board configurations from Diagrams 3-14 to 3-16 (not counting symmetry)
We have determined in Diagram 3-7 that the 2x3 rectangular board (the upper-left board
of Diagram 3-17) allows both L & R to move in Slice 2 on their turns after M, leaving M
with just one available square in that area. This leaves M with only one available space
left in Slice 2 after three turns, so M should avoid placing pieces that will help the
alliance create the 2x3 rectangular area.
The reader can verify that for all the other board configurations, there exists at least one
M move that allows only one alliance member to move in Slice 2 on his next turn, forcing
the other alliance member to play on either Slice 1 or Slice 3. Thus, this leaves M with
three available spaces in Slice 2 after three turns.
This completes the proof of Proposition 3-4.
67
••
3.3 – Analysis of L-M-R and R-M-L Games
In this section, we continue the proofs of Propositions 3-1 and 3-1 by proving the
following for 4x4x3 Domineering games where M plays on the 3 component:
Proposition 3-5: In L-M-R and R-M-L turn order games, the alliance can limit the
number of M’s available spaces in Slice 2 to a maximum of three after 2.75 turns.
Diagrams 3-18a and 3-18b show examples of L and R’s first moves in an L-M-R and an
R-M-L game respectively.
M to play after 0.25 turns M to play after 0.25 turns
•••
Slice 2 (L-M-R game) Slice 2 (R-M-L game)
Diagram 3-18a Diagram 3-18b L’s first move R’s first move
By the end of this chapter, we will show that these are a good first move for the alliance
because M will not be able to force a win. The following diagrams in this section will
follow an L-M-R game, but they can also correlate to a R-M-L game because of
rotational symmetry (the reader can confirm this by looking at Diagrams 3-18a and 3-
18b).
There are fourteen possible moves for M to play in his next turn, but we can reduce this
analysis to seven combinations.
68
R to play after 0.75 turns on each board (all represent Slice 2)
••
••
••
••
Game 1• Game 2• Game 3• Game 4•
••
••
••
Game 5• Game 6• Game 7•
Diagram 3-19 The shaded squares in each diagram demonstrate potential spaces where M can play.
Each Slice 2 in Diagram 3-19 has two shaded squares. These shaded squares represent
spaces where M can play in his next turn. The reason they are shaded is because L and R
are going to agree not to play on those spaces, essentially playing “one-hand tied”. We
will show that the alliance can still prevent M from winning using this self-imposed
handicap. Note that because the alliance has agreed not to play on the shaded squares, we
will consider one of the two shaded squares as a reserved space for M.
M to play after 1.25 turns on each board (all represent Slice 2)
• • ••• •
• •• •• •
••
••• •
••
•• ••
Game 1• Game 2• Game 3 Game 4
• ••
•
••
• •• •
••
• ••
•
••
Game 5• Game 6• Game 7•
Diagram 3-20 From Diagram 3-19, the alliance members make their next move,
agreeing not to play on the shaded squares of each game.
Each game in Diagram 3-20 shows how the alliance can react to M’s move in either
shaded square. We will show that all the moves made in each game are a winning
69
•• ••
••
••
••
••••••
strategy for L and R. Observe in the seven games that there are three possible board
configurations left on Slice 2 (excluding symmetry).
M to play after 1.25 turns on each board configuration
Diagram 3-21a Diagram 3-21b Diagram 3-21c
These diagrams are the remaining spaces from the seven games in Diagram 3-20 (excluding symmetry)
We will show that for all possible board configurations in Diagrams 3-21a, 3-21b, and 3-
21c, M can additionally reserve a maximum of two spaces after 2.75 turns. Let’s start
with Diagram 3-21a.
M to play after 2.25 turns on each board (all represent Slice 2)
M
•• •
• M
• •
•
M ••
M
•• •
• M
• •
•
••• M
•
•
M •
•
•
M •
Diagram 3-22 These boards showcase the eight possible moves M can play
(shaded in the above diagrams) after Diagram 3-21a. Also shown is the alliance’s response to each shaded move.
Diagram 3-22 shows all eight possible moves that M can make from Diagram 3-21a.
Also shown is how the alliance should respond to those shaded moves in each board (we
will show later in this chapter that the alliance’s response in each diagram is a winning
strategy for the alliance).
70
Note that after 2.25 turns, there are three spaces left in every board with M to play next.
If M were to play on the corner of the remaining spaces on each board, it would give M
two reserved spaces and prevent the alliance from placing any more pieces on Slice 2.
We established earlier that one of the shaded squares in Diagram 3-20 was a reserved
space for M, so the maximum number of reserved spaces for M in Diagram 3-21a is
three.
Let’s now examine the possible moves that can be made in Diagram 3-21b.
M to play after 2.25 turns on each board (all represent Slice 2)
M ••
•••
M ••
•••
• M
•
•••
•
M •
•••
•
M •
•••
•• M
•••
•••••
M
•••••
M
Diagram 3-23 These boards showcase the eight possible moves M can play
(shaded in the above diagrams) from Diagram 3-21b. Also shown is the alliance’s response to each shaded move.
Diagram 3-23 shows all eight possible moves that M can make from Diagram 3-21b.
Also shown is how the alliance should respond to those shaded moves in each board (we
will show later in this chapter that the alliance’s response in each diagram is a winning
strategy for the alliance).
Note that after 2.25 turns, there are three spaces left in every board with M to play next.
In the first seven board configurations of Diagram 3-23, if M were to play on the corner
of the remaining spaces on each board, it would give M two reserved spaces and prevent
the alliance from placing any more pieces on Slice 2. In the eighth board of Diagram 3-
23, if M’s next move is not to the left of the shaded M square, then it would give M two
reserved spaces and prevent the alliance from placing any more pieces on Slice 2. We
established earlier that one of the shaded squares in Diagram 3-20 was a reserved space
for M, so the maximum number of reserved spaces for M in Diagram 3-21b is three.
71
Let’s now examine the possible moves that can be made in Diagram 3-21c.
M to play after 2.25 turns on each board (all represent Slice 2)
M
••
•••
M
•
••••
••
M
•••
••
M
•••
••
•••
M
••
M •••
••
•••M
••
••• M
Diagram 3-24 These boards showcase the eight possible moves M can play
(shaded in the above diagrams) from Diagram 3-21c. Also shown is the alliance’s response to each shaded move.
Diagram 3-24 shows all eight possible moves that M can make from Diagram 3-21c.
Also shown is how the alliance should respond to those shaded moves in each board (we
will show later in this chapter that the alliance’s response in each diagram is a winning
strategy for the alliance).
Note that after 2.25 turns, there are three spaces left in every board with M to play next.
The reader can verify that M can play a piece in each of the diagram’s remaining spaces
that would give M two reserved spaces and prevent the alliance from placing any more
pieces on Slice 2. We established earlier that one of the shaded squares in Diagram 3-20
was a reserved space for M, so the maximum number of reserved spaces for M in
Diagram 3-21c is three.
We have now shown that after 2.75 turns, the alliance can hold M to a maximum of three
reserved spaces in Slice 2 for L-M-R and R-M-L turn-order games.
This completes the proof of Proposition 3-5.
72
We have examined L-R-M, R-L-M, M-L-R, M-R-L, L-M-R, and R-M-L turn-order
games to where the alliance can no longer move on Slice 2. We have now established
that the alliance can hold M to a maximum of three reserved spaces in Slice 2 after three
turns, regardless of the turn-order.
Thus, the proofs of Propositions 3-3, 3-4, and 3-5 have proven Proposition 3-2.
3.4 – Endgame Analysis
We will conclude the proof of Proposition 3-1 in this section. In the previous sections,
we showed that the alliance can hold M to a maximum of three reserved spaces in Slice 2
after three turns. It is possible that even if the alliance members do not cooperatively
block any of M’s available moves, they will still have at least one move after M plays on
his reserved spaces in Slice 2. However, there are some situations were the alliance must
cooperatively block at least one of M’s potential moves in order to prevent M from
winning. First, let’s observe how M’s three reserved spaces in Slice 2 affect the endgame
of a 4x4x3 Domineering game.
Result: M to play after 3 turns
•
•
•
•
•
•
Slice 1 Slice 3 Diagram 3-25
The •’s represent potential moves where M can play during his third turn.
Diagram 3-25 shows potential areas in Slices 1 and 3 that M’s piece would cover
(marked with •’s) if he played on his reserved spaces in Slice 2. We are first going to
show that if the alliance agrees not to play on any of the potential squares that M can
play, each alliance member can place a minimum of five pieces on each slice, as long as
one member plays only on one slice and the other only plays on the other slice (for
example, if L plays on Slice 1, then R plays on Slice 3).
73
Lemma: For any arrangement of three •’s in corresponding spaces on Slice 1 and Slice
3, we can place at least five L pieces on one slice and at least five R pieces on the other.
Proof:
Note: Diagrams 3-26a to 3-26c refer to Slice A and Diagrams 3-27a to 3-27c refer to
Slice B, where either A = 1 and B = 3, or A = 3 and B = 1.
Case 1: On a slice with •’s on three distinct columns, we can place two L pieces on the
blank column and one L piece on the other columns. That gives us a total of five L
pieces. See Diagram 3-26a for an example.
Case 2: On a slice with •’s that are not on three distinct columns, there are at least two
columns that are blank (two L pieces per empty column), and at least one that has less
than or equal to one • (at least one L piece on this column). This gives us a total of at
least five L pieces. See Diagrams 3-26b and 3-26c for examples.
Case 1 example Case 2 example Another Case 2 example
• • •
• • •
• • • •• • •
• • •
• • •
• • • •• • •
••••••••
••••••••
• ••••••
••••••••
Potential Slice A Potential Slice A Potential Slice A
Diagram 3-26a Diagram 3-26b Diagram 3-26c The •’s are in three The •’s are not in three The •’s are on the same
distinct columns. L is able distinct columns. L is able column. L is able to to place five pieces in this to place five pieces in this place six pieces in this
configuration configuration. configuration.
Case 3: On a slice with •’s on three distinct rows, we can place two R pieces on the
blank row and one R piece on the other rows. That gives us a total of five R pieces. See
Diagram 3-27a for an example.
Case 4: On a slice with •’s that are not on three distinct rows, there are at least two rows
that are blank (two R pieces per empty row), and at least one that has less than or equal to
74
one • (at least one R piece on this row). This also gives us a total of at least five R
pieces. See Diagrams 3-27b and 3-27c for examples. ••
Case 3 example Case 4 example Another Case 4 example
• ••
• ••
•• •
•• ••
• ••
• •
•• ••
•• ••
•••• •••••• •••
••• •••
••• •••
Potential Slice B Potential Slice B Potential Slice B
Diagram 3-27a The •’s are in three
Diagram 3-27b The •’s are not in three
Diagram 3-27c The •’s are on the same
distinct rows. R is able to distinct rows. R is able to row. R is able to place six place five pieces in this place five pieces in this pieces in this configuration.
configuration configuration.
Although we showed that L and R can each place at least five pieces on their respective
slices regardless of how M’s three potential spaces are configured on the board, we did
not take into account the three M pieces that are already on the board. Recall that we
proved that the alliance can limit M to a maximum of three reserved spaces on Slice 2
after three turns. After three complete turns, M has moved three times; thus, there are
three M pieces on the board. Those three pieces will affect how many pieces each
alliance member can play on each slice, as demonstrated in the following diagram.
M M
• •
• • • •• • •
••• •••
• •
•• M
••• •••
Slice A Slice B
Diagram 3-28 The three M pieces affect how many pieces each alliance member can play on each slice.
Diagram 3-28 represents a potential 4x4x3 game after three turns. The two M pieces on
Slice A reduce L’s five potential moves that will not cover a potential M move to three,
75
and the one M piece on Slice B reduces R’s five potential moves that will not cover a
potential M move to four.
Observation: Each M piece on a slice reduces the number of an alliance member’s
potential moves (one that doesn’t block •) on that slice.
The reader may wonder why the alliance is agreeing to play with “one-hand tied” by not
cooperatively covering M’s remaining spaces. If we can show that the alliance still has
one move left after M’s final move even if it does not cover M’s remaining moves, then
we have definitively proven that the alliance can prevent M from winning. Next, we will
show the effectiveness of this “one-hand tied” strategy in M-L-R and M-R-L games.
In Section 3.2, we showed in Diagram 3-17 that after the third turn in an M-L-R and M-
R-L game, M has three reserved spaces on Slice 2 and only one of the alliance members
was able to move on Slice 2. This forces the other alliance member to play on either
Slice 1 or Slice 3 in his third turn. Where that alliance member should play in his third
turn depends on how many M pieces are already on Slices 1 and 3. Since there are three
M pieces on the board after three turns, there will either be no M pieces on Slice A and
three M pieces on Slice B, or there will be one M piece on Slice A and two M pieces on
Slice B (where A = 1 and B = 3, or A = 3 and B = 1). The alliance member’s strategy is
as follows:
Strategy: The alliance member forced to play on either Slice 1 or 3 should choose the
slice that has the least number of M pieces and keep playing on that slice for the rest of
the game until a move there is no longer available.
For example, let’s say R cannot place a piece on Slice 2 during his third turn. The board
has one M piece on Slice 1 and two M pieces on Slice 2. Thus, R chooses to play on
Slice 1 in his third turn, and will keep playing on that slice for the rest of the game. The
other alliance member should not play on the same slice where his partner played,
because such a move would block two potential moves for his partner.
Strategy: The other alliance member should play on the slice that has more M pieces for
the rest of the game until a move is no longer available.
76
In our previous example, R chose to play on Slice 1 because it has fewer M pieces.
Therefore, L needs to play on Slice 3 for the rest of the game to avoid overlapping R’s
potential moves on Slice 1 unless there are no more moves available on Slice 3.
In our analysis of M-L-R and M-R-L games, there are two possible scenarios to examine:
• In games with no M pieces on one slice and three M pieces on the other, the
player who cannot play on Slice 2 in his third turn plays on the slice with no M
pieces.
• In games with one M piece on one slice and two M pieces on the other, the player
who cannot play on Slice 2 in his third turn plays on a slice with one M piece.
We will start with the first case for M-L-R and M-R-L games, where an alliance member
is forced to play on a Slice 1 or 3 with no M pieces.
Number of remaining moves in each turn of M-L-R & M-R-L games
Turn Number
M-L-R Game: Who Plays Next
M-R-L Game: Who Plays Next
Number of Remaining Moves
For M For 2nd Player For 3rd Player For Alliance
3 M to play next M to play next 3 2 4 6
3.5 L to play next R to play next 2 2 4 6
3.75 R to play next L to play next 2 1 4 5
4 M to play next M to play next 2 1 3 4 4.5 L to play next R to play next 1 1 3 4
4.75 R to play next L to play next 1 0 3 3 5 M to play next M to play next 1 0 2 2
5.5 L to play next R to play next 0 0 2 2
Diagram 3-29
Diagram 3-29 reflects 4x4x3 M-L-R and M-R-L games where the third player is forced to
play on either Slice 1 or 3 that has no M pieces during the third turn. The second player
will play on the slice that has three M pieces in his next turn, and both alliance members
will play on their respective slices (for example, if L plays on Slice 1, then R plays on
Slice 3) for the rest of the game unless a move is no longer available on that slice. The
third player had five available moves on the slice with no M pieces, but because he was
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forced to play on that slice during the third turn, his five available moves are reduced to
four. The second player will play on the other slice that has three M pieces, so his five
available moves are reduced to two. The reader will note in the chart that during the fifth
turn, when M plays on his last reserved space, the third player has two moves left in his
slice. Since an alliance member has at least one move after M played his final piece, this
is a loss for M.
This chart presumes that the third player of M-L-R and M-R-L games was forced to play
on Slice 1 or 3 with no M pieces during his third turn. If the second player was forced to
play on Slice 1 or 3 with no M pieces during his third turn, the total number of remaining
moves for the alliance would remain the same after each turn, as indicated in the
following chart.
Number of remaining moves in each turn of M-L-R & M-R-L games
Turn Number
M-L-R Game: Who Plays Next
M-R-L Game: Who Plays Next
Number of Remaining Moves
For M For 2nd Player For 3rd Player For Alliance
3 M to play next M to play next 3 4 2 6 3.5 L to play next R to play next 2 4 2 6
3.75 R to play next L to play next 2 3 2 5 4 M to play next M to play next 2 3 1 4
4.5 L to play next R to play next 1 3 1 4 4.75 R to play next L to play next 1 2 1 3
5 M to play next M to play next 1 2 0 2
5.5 L to play next R to play next 0 2 0 2
Diagram 3-30
The chart in Diagram 3-30 shows that the alliance has at least one more move after M’s
final move, which is a loss for M.
Next, we will analyze the second case for M-L-R and M-R-L games, where an alliance
member is forced to play on a Slice 1 or 3 with one M piece.
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Number of remaining moves in each turn of M-L-R & M-R-L games
Turn Number
M-L-R Game: Who Plays Next
M-R-L Game: Who Plays Next
Number of Remaining Moves
For M For 2nd Player For 3rd Player For Alliance
3 M to play next M to play next 3 3 3 6
3.5 L to play next R to play next 2 3 3 6
3.75 R to play next L to play next 2 2 3 5
4 M to play next M to play next 2 2 2 4 4.5 L to play next R to play next 1 2 2 4
4.75 R to play next L to play next 1 1 2 3 5 M to play next M to play next 1 1 1 2
5.5 L to play next R to play next 0 1 1 2
Diagram 3-31
Diagram 3-31 reflects 4x4x3 M-L-R and M-R-L games where the either the second or
third player is forced to play on either Slice 1 or 3 that has one M piece during the third
turn. The other alliance member will play on the other slice that has two M pieces in his
next turn. Both alliance members will then play on their respective slices (for example, if
L plays on Slice 1, then R plays on Slice 3) for the rest of the game unless a move is no
longer available on that slice. The alliance member forced to play on the slice with one
M piece had four available moves, but because he was forced to play on that slice during
the third turn, his four available moves were reduced to three. The other alliance member
will play on the other slice that has two M pieces, so his five available moves are reduced
to three. The reader will note that during the fifth turn, when M plays on his last reserved
space, both alliance members have one move left. Since the alliance has at least one
move after M played his final piece, this is a loss for M.
We have shown that the alliance can prevent M from winning all possible M-L-R and M-
R-L games, even if they play “one-hand tied” and do not cooperatively cover any of M’s
three remaining moves. We will next discuss how the alliance can prevent M from
winning L-M-R and R-M-L games.
Unfortunately, we cannot apply the “one-hand tied” strategy for L-M-R and R-M-L
games to show that the alliance can prevent M from winning. In Section 3.3, we showed
in Diagrams 3-22, 3-23, and 3-24 that after 2.75 turns, M will place his third piece on
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Slice 2, which prevents both L and R to play any more pieces on Slice 2. If the alliance
members were to ignore the opportunity to cooperatively block M in their next turn,
neither will be able to play a piece after M plays his last piece, which results in a win for
M.
In our analysis of L-M-R and R-M-L games, there are two possible scenarios to examine:
• In games with no M pieces on one slice and three M pieces on the other, the third
player plays on the slice with no M pieces in his third turn.
• In games with one M piece on one slice and two M pieces on the other, the third
player plays on the slice with one M piece in his third turn.
We will start with the first case for L-M-R and R-M-L games, where the third player is
forced to play on a Slice 1 or 3 with no M pieces.
Number of available moves for each turn in L-M-R & R-M-L game
Turn Number
L-M-R Game: Who Plays Next
R-M-L Game: Who Plays Next
Number of Remaining Moves
For M For 1st Player For 3rd Player For Alliance
2.75 R to play next L to play next 3 2 5 7 3 L to play next R to play next 3 2 4 6
3.25 M to play next M to play next 3 1 4 5
3.75 R to play next L to play next 2 1 4 5 4 L to play next R to play next 2 1 3 4
4.25 M to play next M to play next 2 0 3 3 4.75 R to play next L to play next 1 0 3 2
5 L to play next R to play next 1 0 2 2
5.25 M to play next M to play next 1 0 0 0
Diagram 3-32
Diagram 3-32 presumes that there are three M pieces on one slice and the third player
decided to play on a slice that has no M pieces. The third player has five potential
moves, as his slice has no M pieces, and the first player has two potential moves, as his
slice has three M pieces. The problem for the alliance occurs after 4.75 turns. After the
third player plays on his slice, there are two potential moves left. However, the first
player has no moves left on his respective slice, so he is forced to play on the other slice,
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which will overlap the third player’s remaining two moves. Since neither alliance
member has any moves left after M plays his final piece, this game is a win for M.
Therefore, the alliance must cooperatively block at least one of M’s moves to prevent M
from winning in L-M-R and R-M-L games.
A reserved move for M is one that neither L nor R can block at any point during the
game. M can protect a potential move (and thus reserve it) if he places pieces around the
potential move in such a way that neither alliance member can overlap it. We are going
to prove the following to show that the alliance can block at least one of M’s potential
moves:
Proposition 3-6: M cannot protect all of his three potential moves with his three M
pieces on a 4x4 slice.
Proof:
We previously showed that, after 2.75 turns in L-M-R and R-M-L games, M will have
three reserved spaces in Slice 2 and three M pieces on the board. There exists no
configuration on a 4x4 slice where M’s three pieces will protect all three of his potential
moves. This is because in order for M to protect one of his moves on either Slice 1 or
Slice 3, he needs at least two pieces.
• M• • •
M•••M••
M• •
• •
M • M
M
• M
M
Potential Slice 1 or 3 Potential Slice 1 or 3 Potential Slice 1 or 3
Diagram 3-33a Diagram 3-33b Diagram 3-33c •’s in center spaces need
four M pieces to be •’s in spaces that are
neither corner nor center •’s in corner spaces need
only two M pieces to be protected need three M pieces to be protected
protected
Diagrams 3-33a to 3-33c show how many M pieces are needed to protect a potential
move depending on its location. Diagram 3-33a is not possible because we stated that M
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only has three pieces on the board. Diagram 3-33b has all of M’s three pieces protecting
only one reserved space, but this leaves the other potential moves unprotected and can be
covered by an alliance member. Diagram 3-33c is the best of the three diagrams because
M protected a potential move with just two M pieces. However, if we were to place
another • on Diagram 3-33c, M cannot protect it with the third M piece because we
showed using Diagrams 3-33a to 3-33c that M needs at least two pieces to protect a
reserved space. If M cannot protect two potential moves with his three pieces, he
certainly cannot protect three potential moves with his three pieces. Therefore, we can
confidently state that the alliance can cooperatively block one of M’s potential moves in
its next turn. •
This concludes the proof of Proposition 3-6.
Let’s reexamine the first case for L-M-R and R-M-L games, where the third player is
forced to play on a Slice 1 or 3 with no M pieces, and have the alliance cooperatively
block one of M’s available moves.
Number of available moves for each turn in L-M-R & R-M-L game
Turn Number
L-M-R Game: Who Plays Next
R-M-L Game: Who Plays Next
Number of Remaining Moves
For M For 1st Player For 3rd Player For Alliance
2.75 R to play next L to play next 3 2 5 7 3 L to play next R to play next 3 2 4 6
3.25 M to play next M to play next 2 1 4 5
3.75 R to play next L to play next 1 1 4 5 4 L to play next R to play next 1 1 3 4
4.25 M to play next M to play next 1 0 3 3 4.75 R to play next L to play next 0 0 3 3
Diagram 3-34
Diagram 3-34 presumes that there are three M pieces on one slice and the third player
plays on the slice with no M pieces. We will assume that the alliance’s attempt to
cooperatively block one of M’s moves will reduce each member's available moves on
each slice by one. After 2.75 turns, the third player has five potential moves, as his slice
has no M pieces, and the first player has two potential moves, as his slice has three M
pieces. M will have two available moves because the alliance will cooperatively block
one of his three remaining moves in Turns 2.75 and 3. One can see that after 4.75 turns,
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the alliance has at least one more move after M plays his final piece, resulting in a loss
for M.
Next, we will analyze the second case for L-M-R and R-M-L games, where the third
player is forced to play on a Slice 1 or 3 with one M piece.
Number of available moves for each turn in L-M-R & R-M-L game
Turn Number
L-M-R Game: Who Plays Next
R-M-L Game: Who Plays Next
Number of Remaining Moves
For M For 1st Player For 3rd Player For Alliance
2.75 R to play next L to play next 3 3 4 7 3 L to play next R to play next 3 3 3 6
3.25 M to play next M to play next 2 2 3 5 3.75 R to play next L to play next 1 2 3 5
4 L to play next R to play next 1 2 2 4
4.25 M to play next M to play next 1 1 2 3 4.75 R to play next L to play next 0 1 2 3
Diagram 3-35
Diagram 3-35 presumes that there are two M pieces on one slice and the third player
plays on the slice with one M piece. After 2.75 turns, the third player has four potential
moves, as his slice has one M piece, and the first player has three potential moves, as his
slice has two M pieces. M will have two available moves because the alliance will
cooperatively block one of his three remaining moves in Turns 2.75 and 3. The total
number of alliance moves for each turn in this chart is the same as those listed on
Diagram 3-34. The alliance has at least one more move after M plays his last piece,
which results in a loss for M.
We have demonstrated that the alliance can prevent M from winning all possible L-M-R
and R-M-L games as long as they cooperatively block at least one of M’s three available
moves. We will next discuss how the alliance can prevent M from winning L-R-M and
R-L-M games.
For L-R-M and R-L-M games, we demonstrated in Section 3.2 that the alliance can limit
M to two reserved spaces on Slice 2 after 2.5 turns. This means that after M plays on his
third turn, he will only have one move left. The alliance has a least two potential moves
in each slice, since one slice could have up to three M pieces, which would reduce an
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alliance member’s five potential moves to two. Thus, even if the alliance members
decide not to cooperatively block M’s last remaining move, they are guaranteed to move
after M plays his final piece, which results in a loss for M.
Therefore, we have shown that for all possible games on a 4x4x3 board, where M plays
on the 3 component, M cannot force a win if the alliance teams up against him and uses
the strategies we described in this chapter.
This completes the proof of Proposition 3-1.
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4.0 – Strategies to Solve a 4x4x3 Domineering Game: Part 2
In the previous chapter, we proved that the alliance can prevent M from winning a 4x4x3
game where M plays on the 3 component (see Chapter 3 for an explanation of what it
means to play on a component). To complete the proof that a 4x4x3 game is solved to be
a game, we are going to prove the following in this chapter:
Proposition 4-1: M is unable to force a win in a 4x3x4 game, where M plays on the 4
component, if the L & R alliance teams up against him.
4.1 – Part 1 of Alliance’s Winning Strategy
The first part of L & R’s winning strategy against M is similar to the first part of the
alliance’s winning strategy against M in 3x3x3 and 4x4x3 (where M plays on the 3
component) games, which is to cover as many spaces of Slice 2 as possible. However,
because M is able to move on four slices instead of three, the alliance must now cover
two slices of the game board to block M from making any more moves. We will show
later in this chapter which two slices are most effective to stopping M.
The second part of the alliance’s winning strategy is also similar to the second part of the
alliance’s winning strategy we covered in previous chapters, which is that the alliance
should cover the corresponding squares above and below any remaining M playable
spaces to block M from playing on those squares. However, there are certain situations
in the winning strategy where the alliance is only able to block above but not below or
below but not above remaining M playable moves.
Throughout this chapter, we will use the phrase “L & R should play…” in order to
describe the strategy L & R use to prevent M from winning,
To assist in the analysis of a 4x3x4 game (where M plays on the 4 component), we
created an application that simulates three players playing a 4x3x4 Domineering game.
This application was developed because this type of analysis contains too many game
tree branches to review by hand. The application considers all possible moves M can
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make during a game and uses a heuristic to determine moves L and R can make in
response to each of M’s possible moves. The goal of the heuristic is to pick moves for
the alliance that will lead to its victory.
This heuristic differs from the winning strategies devised in the previous chapters such
that not every move in the heuristic is necessarily a best-play move for the alliance. This
is because the heuristic was designed to be generic enough to avoid writing special-case
responses to a multitude of specific scenarios. The reader should be aware that even if a
move made by the heuristic does not appear to be a best-play move for the alliance, it is
still designed to be a winning move for the alliance.
The first part of the alliance’s winning strategy is as follows:
Strategy: Cover as much of Slices 1 & 3 as possible for the duration of the game until L
and R are unable to move on either slice.
It is crucial that the alliance commits to playing on either Slices 1 or 3 until they cannot
move on those slices anymore. Deciding which of the two slices to play will depend on
where M last played.
M will play on Slices j and j+1 in his turns, where j = 1, 2, or 3. If j = 1 or 3, the alliance
should play on the same Slice j. If j = 2, then the alliance should play on Slice 3.
However, if an alliance member is unable to play on one of the slices in the Slices 1 & 3
pair, then he should move on the other slice in the pair if a move is available. For
example, let’s say M played on Slices 1 and 2. According to our strategy, the alliance
should move on Slice 1 on its next turn. However, if L or R is unable to move on Slice 1,
then that alliance member should play on Slice 3 if a move is available. If an alliance
member is unable to play on either Slice 1 or Slice 3, then he must follow the second part
of the alliance’s winning strategy, which we will cover in the next section.
In the following pages, we will explain the coordinate system of our 3D Domineering
simulation application so that we can accurately describe where L and R should play on
Slices 1 and 3.
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In the application, a move is represented by a three-dimensional coordinate, (x, y, z),
where x represents the column of a slice, y represents the row of a slice, and z represents
the actual slice. For example, a coordinate that is labeled (0, 0, 0) represents the space
located at the first column and first row of the first slice in a 3D Domineering game. The
coordinate system in the application follows a zero-based index, which is why the first
column, first row, first slice of a 3D Domineering game is (0, 0, 0) and not (1, 1, 1). The
following diagram demonstrates a 3x3x3 Domineering game with each space labeled
using our 3D Domineering coordinate system.
A 3x3x3 Domineering Game
0,0,0 1,0,0 2,0,0
0,1,0 1,1,0 2,1,0
0,2,0 1,2,0 2,2,0
0,0,1 1,0,1 2,0,1
0,1,1 1,1,1 2,1,1
0,2,1 1,2,1 2,2,1
0,0,2 1,0,2 2,0,2
0,1,2 1,1,2 2,1,2
0,2,2 1,2,2 2,2,2
Slice 1 Slice 2 Slice 3
Diagram 4-1 Each square is labeled with the 3D Domineering coordinate system used by the simulation
application. The shaded squares indicate a possible move made by L, R, and M.
You may have noticed that while a move is represented by one three-dimensional
coordinate, a piece in a 3D Domineering game takes two spaces. The application is able
to identify the second half of a move based on the player who made the move. For
example, if L makes a move at (0, 0, 0), the application knows that the second-half of that
move is on (0, 1, 0). If R makes a move at (0, 1, 1), the application knows that the
second-half is on (1, 1, 1). And if M makes a move in (2, 2, 1), the application knows
that the second-half is on (2, 2, 2) (see shaded squares in Diagram 4-2). Keep this in
mind as we describe later in this chapter what moves the alliance members should make
for their winning strategy.
Now that we explained how a move is described using our application’s 3D Domineering
coordinate system, let us examine where L and R should play on a playable slice (either
Slice 1 or Slice 3) using the following chart.
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Alliance’s Winning Strategy Details and How They Depend on the Number of Pieces in a Given Slice
Number of pieces on alliance's next playable slice (Slice 1 or 3)
Special condition The moves L & R should make as
part of the alliance’s winning strategy.
# of L pieces
# of R pieces
# of M pieces
L’s move R’s move
0 0 0 None (0, 1, 1 or 3) (0, 0, 1 or 3)
0 0 1 M piece is on Column 3 or 4 (0, 1, 1 or 3) (0, 0, 1 or 3)
0 0 1 M piece is on Column 1 or 2 (3, 0, 1 or 3) (2, 2, 1 or 3)
1 0 0 None (3, 0, 1 or 3) (2, 2, 1 or 3)
0 1 0 None (3, 0, 1 or 3) (2, 2, 1 or 3)
1 1 0 None (2, 0, 1 or 3) (3, 1, 1 or 3)
1 0 1 M piece is on Column 3 or 4 (1, 1, 1 or 3) (0, 0, 1 or 3)
1 0 1 M piece is on Column 1 or 2 (3, 0, 1 or 3) (2, 2, 1 or 3)
0 1 1 M piece is on Column 3 or 4,
Row 1 (0, 1, 1 or 3) (2, 2, 1 or 3)
0 1 1 M piece is on Column 3 or 4,
but not Row 1 (0, 1, 1 or 3) (2, 0, 1 or 3)
0 1 1 M piece is on Column 1 or 2 (3, 0, 1 or 3) (2, 2, 1 or 3)
1 1 1 See Diagrams 4-10 and 4-11
2 1 0 None (2, 0, 1 or 3) (0, 0, 1 or 3)
1 2 0 None (2, 0, 1 or 3) (0, 1, 1 or 3)
2 2 0 None First player of alliance moves on
slice. Other must play on a different slice.
1 1 2 None First player of alliance plays move on slice that won’t block second
player from moving on same slice.
Total >= 5 None
If both players cannot move on same slice, then first player who can move on slice should take it
while other player plays on different slice. If no move is
available, then both members play on a different slice.
Diagram 4-2 This chart summarizes the moves the alliance should make on Slice 1 or 3 as part of its winning strategy.
The chart in Diagram 4-2 summarizes the first part of the alliance’s winning strategy. It
shows which moves the alliance should make when they can play on Slice 1 or 3. Those
moves are dependent on how many pieces are already on that slice and, in some cases,
where those pieces are located. We will describe the alliance’s moves demonstrated in
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••
this chart with more detail in the following pages, starting with the alliance playing on an
empty slice.
1) Alliance plays on an empty slice
There are two situations when the alliance members will play on an empty slice. The
first situation is at the start of an L-R-M, R-L-M, L-M-R or R-M-L game. The following
demonstrates where the alliance should play at the start of those games.
First moves L and R should make - M to play next
L-R-M or R-L-M Game L-M-R Game R-M-L Game
••
••
••
Slice 1 Slice 1 Slice 1
Diagram 4-3a Diagram 4-3b Diagram 4-3c The first moves L and R The first move L should The first move R should
should make in this game. make in this game. make in this game.
The alliance’s winning strategy relies on where M last played, but since M has not moved
yet in this particular example, we picked Slice 1 as the place where the alliance will first
move. Note that R moved on the upper-left corner of the slice and L moved on the first
column below R’s piece in their turns. These moves (indicated in Diagrams 4-3a, 4-3b,
and 4-3c) are where the alliance members should play if they encounter an empty slice.
The next example demonstrates the second situation where the alliance will play on an
empty slice. There are certain 4x3x4 games where M moves on one of the slices of the
Slice 1 & 3 pair that neither alliance member can move on in their next turn, and the
other slice in the pair happens to be empty. The following diagram demonstrates a
potential 4x3x4 game where such a scenario can occur.
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••
•••• ••
A sample 4x3x4 game (M-L-R/M-R-L) M to play after three turns
•• M •
• M •
• •• M
M
M
M
••
••
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-4 On M’s third turn, M plays on Slice 1.
L and R can’t move on Slice 1, so they move on Slice 3, which is empty.
In Diagram 4-4, M has made three moves on Slices 1 and 2. L and R have been able to
move on Slice 1 in their first two turns, but are unable to play on that slice again in their
third turn. Therefore, according to our winning strategy, the alliance must play on Slice
3, which M has not played in this sample game. Note that the moves the alliance played
on Slice 3 are the ones we indicated in Diagram 4-3a.
2) Alliance plays on a slice with one piece
If the alliance were to play on Slice 1 or 3 that already has one M piece, then the
following demonstrates the winning strategy for the alliance.
M to play next
• ••
• • •
• • •
••
•
Diagram 4-5a Diagram 4-5b Either Slice 1 or 3 Either Slice 1 or 3
If M plays on any of the shaded spaces of an empty slice (Slice 1 or 3), the alliance should respond accordingly.
If M played on a previously empty slice (either Slice 1 or 3), the alliance’s response will
depend on what column M’s piece is on. If M plays on the third or fourth row of the
slice, then R should play on the upper-left corner of the slice and L should play a piece
below it on the first column (as shown in Diagram 4-5a). If M plays on the first or
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second column of the slice, then R should play on the lower-right corner of the slice and
L should play a piece above it on the fourth column (as shown in Diagram 4-5b).
The reader will notice that L and R play their pieces away from where M last played his
piece in Diagrams 4-5a and 4-5b. The primary reason is that it permits the alliance to
place their pieces along the edge of a slice. This prevents M from trying to reserve
spaces on the corners of the slice.
If the alliance were to play on Slice 1 or 3 that already has one L piece or one R piece,
then the following demonstrates the winning strategy for the alliance.
M to play next
•
••
• ••
•• ••
••
Diagram 4-6a Diagram 4-6b Either Slice 1 or 3 Either Slice 1 or 3
The above diagrams show where L and R should play on a Slice 1 or 3 if it already has an L or R piece.
In both diagrams, R plays on the lower right corner of the slice and L plays above it on
the fourth column. We established in Diagrams 4-3b and 4-3c where L and R would play
on an empty slice, so the gray pieces in Diagrams 4-6a and 4-6b reflect those moves.
3) Alliance plays on a slice with two pieces
We will now describe the alliance’s winning strategy if their next playable slice already
has two pieces.
If the alliance were to play on Slice 1 or 3 that already has an L piece and an R piece,
then the following demonstrates the winning strategy for the alliance.
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M to play next
•• ••
• •• •
Either Slice 1 or 3
Diagram 4-7 L and R should play on the following spaces in this scenario
We indicated in Diagram 4-1a where L and R should play if they start on an empty slice,
so the grayed L and R pieces in the above diagram reflect those moves. The alliance’s
next move if they were forced to play in this Slice 1 or 3 is for R to play on the upper-
right corner and L to play below it on the fourth column (see Diagram 4-7).
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on Slice 1 or 3 that already has an L piece and an M piece.
M to play next
••
• •• •
•
••
• ••
Diagram 4-8a Diagram 4-8b Either Slice 1 or 3 Either Slice 1 or 3
If M plays on any of the shaded spaces, the alliance should respond accordingly.
We indicated in Diagram 4-3b where L should play if he starts on an empty slice, so the
grayed L pieces in both Diagrams 4-8a and 4-8b reflect that move. In Diagram 4-8a, the
shaded squares represent M playing on either the third or fourth column, so the alliance’s
winning strategy is for R to play on the upper-left corner, and L to play below that piece
on the second column. In Diagram 4-8b, the shaded squares represent M playing on
either the first or second column, so the alliance’s winning strategy is for R to play on the
lower-right corner and L to play above it on the fourth column.
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on a Slice 1 or 3 that already has an R piece and an M piece.
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M to move next
••
•• ••
•• ••
••
•• ••
••
Diagram 4-9a Diagram 4-9b Diagram 4-9c Either Slice 1 or 3 Either Slice 1 or 3 Either Slice 1 or 3
If M plays on any of the shaded spaces, the alliance should respond accordingly.
We indicated in Diagram 4-3c where R should play if he starts on an empty slice, so the
grayed R pieces in Diagrams 4-9a, 4-9b, and 4-9c reflect that move. In Diagram 4-9a, the
shaded squares represent M playing on the first row (either third or fourth column), so the
alliance’s winning strategy is to respond with R playing on the lower-right corner, and L
playing on the first column. In Diagram 4-9b, the shaded squares represent M playing on
either the third or fourth column that isn’t on the first row, so the alliance’s winning
strategy is to respond with R playing on the upper-right corner and L paying on the first
column. In Diagram 4-9c, the shaded squares represent M playing on the first or second
column, so the alliance’s winning strategy is to respond with R playing on the lower-right
corner and L playing above it on the fourth column.
4) Alliance plays on a slice with three pieces
We will now describe the alliance’s winning strategy if their next playable slice already
has three pieces.
Diagrams 4-10 and 4-11 demonstrate the alliance’s winning strategy if the alliance were
to play on Slice 1 or 3 that already has an L piece, an R piece, and an M piece.
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All boards represent a potential Slice 1 or 3 M to play next on each one
•• ••
• M •• •
•• ••
• M •• •
•• ••
• •• M •
•• ••
• •• M •
•• M •
• •
• ••
•• M
• •• •• •
•• ••
• • M
• •
•• •
• •
• •• M
Diagram 4-10 These boards show how the alliance should respond to the eight
possible moves by M (indicated by shaded square) in this scenario.
In Diagram 4-10, each board has a grayed R piece on the upper-left corner and a grayed L
piece below it on the first column. We indicated in Diagrams 4-3a and 4-5a the scenarios
where the alliance would make those moves. Each board in Diagram 4-10 shows where
the alliance can play in their next turn depending on the location of M’s piece.
All boards represent a potential Slice 1 or 3 M to play next on each one
• M •• •
•• ••
• M •• •
•• ••
• •• M •
•• ••
• •• M •
•• ••
M •• •
• •
• ••
• •
M • •
•• ••
• •• •• •
M ••
•• •
• •
• M ••
Diagram 4-11 These boards show how the alliance should respond to the eight
possible moves by M (indicated by shaded square) in this scenario.
In Diagram 4-11, each board has a grayed R piece on the lower-right corner and a grayed
L piece above it on the fourth column. We indicated in Diagram 4-5b the scenario where
the alliance would make those moves. Each board in Diagram 4-11 shows how the
alliance can play in their next turn depending on the location of M’s piece.
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on Slice 1 or 3 that already has two L pieces and one R piece.
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M to play next
•• • •
•• •
• ••
Diagram 4-12 Either Slice 1 or 3
The above diagram shows where L and R should play, if the above slice already has the grayed piece configuration
The three grayed pieces in Diagram 4-12 were moves that were already indicated in
Diagram 4-6a. The alliance’s winning strategy is for R to play on the upper-left corner
and L to play on the third column.
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on a Slice 1 or 3 that already has two R pieces and one L piece.
M to play next
•• • •
•• • •
••
Diagram 4-13 Either Slice 1 or 3
The above diagram shows where L and R should play, if the above slice already has the grayed piece configuration
The three grayed pieces in Diagram 4-13 were moves that were already indicated in
Diagram 4-6b. The alliance’s winning strategy is for R to play on the second row across
the first and second columns and L to play on the third column.
5) Alliance plays on a slice with four pieces
We will now examine the alliance’s winning strategy if their next playable slice already
has four pieces.
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on a Slice 1 or 3 that already has two L pieces and two R pieces.
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L or R to play next
•• ••
• •• •
Either Slice 1 or 3
Diagram 4-14 In this scenario, only one of the alliance members can move in the available spaces.
The gray pieces in Diagram 4-14 were moves that were already indicated in Diagram 4-7.
Only one of the alliance members is able to move in this slice. The other member will
either have to move on the other slice of the Slice 1 & 3 pair if such a move is available,
or move on Slice 2 or Slice 4. If an alliance member is forced to play on either Slice 2 or
Slice 4, then that member must follow the second part of the alliance’s winning strategy.
That strategy will be explained later in this chapter.
If the alliance’s next playable slice has four pieces and there is at least one M piece on
that slice, then the alliance’s winning strategy will be for one alliance member to perform
his next available move that will not prevent the next alliance member from playing on
the same slice. The following diagram shows an example of this strategy.
Sample 4x3x4 games M to move next
M-L-R Game M-R-L Game
M • •
•• • •
M ••
M •• •
••
•M ••
Diagram 4-15a Diagram 4-15b Either Slice 1 or 3 Either Slice 1 or 3
After M’s turn (indicated by shaded square), an alliance member played the next available move that would not block the other alliance member from playing on the same slice.
Diagrams 4-15a and 4-15b show how the alliance members have played pieces that did
not prevent the other member from playing on the same slice. The reader will notice that
the alliance played different pieces in the two diagrams despite the board configuration
being the same before the alliance moved. These moves are not hard-coded—they are
made based on what moves are available for each player. To understand why these
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different moves were made, we need to explain how the application handles available
moves for each player.
Before the application simulates a 3D Domineering game, it generates all the possible
moves that each player can make. These possible moves are stored in a sorted set that are
sorted by the z coordinate first, then the y coordinate, and finally the x coordinate.
Therefore, (1, 0, 0) is sorted before (0, 1, 0) in the set, and (1, 2, 0) is sorted before (0, 0,
1). Once the simulation starts, every move made by each player removes available
spaces, and those spaces are removed from each player's available moves set if they exist
in the set.
Diagram 4-15a shows a sample M-L-R game where L moves next after M plays on the
second column, third row of the potential Slice 1 or 3 (indicated by the shaded square).
Space (1, 0, 1 or 3) is the first available move in L's sorted set of available moves that L
can play on the potential Slice 1 or 3. However, if L were to play on that space, R would
not be able to play on that same slice. Since that goes against the alliance’s winning
strategy, we will need to find another available for L that would still allow R to play on
the same slice. L's next available move on the potential Slice 1 or 3 in his available
moves set is (2, 0, 1 or 3), which is an acceptable move according to the alliance’s
winning strategy because it allows R to move on the same slice at (0, 1, 1 or 3).
Diagram 4-15b shows a sample M-R-L game where R moves next after M plays on the
second column, third row of the potential Slice 1 or 3 (indicated by the shaded square).
Space (1, 0, 1 or 3) is the first available move in R’s sorted set of available moves that R
can play on the potential Slice 1 or 3. This is an acceptable move according to the
alliance’s winning strategy because it allows L to move on that same slice at (0, 1, 1 or
3).
Although the alliance performed different moves when the turn order of the game
differed, they are still winning moves because they accomplish the same goal—to cover
as many spaces on Slices 1 and 3 as possible to remove playable spaces for M.
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6) Alliance plays on a slice with five pieces
We will now examine the alliance’s winning strategy if their next playable slice already
has five pieces.
The following diagram demonstrates the alliance’s winning strategy if the alliance were
to play on a Slice 1 or 3 that already has two L pieces, two R pieces, and one M piece.
All boards represent a potential Slice 1 or 3 L or R to play next on each one
•• ••
• M •• •
•• ••
• M •• •
•• ••
• •• M •
•• ••
• •• M •
•• M •
• •
• ••
•• M
• •• •• •
•• ••
• • M
• •
•• •
• •
• •• M
Diagram 4-16 In these scenarios, only one of the alliance members can move in the available spaces.
Diagram 4-16 contain possible board configurations of a potential Slice 1 or 3 with two L
pieces, two R pieces, and one M piece that were already indicated by moves made in
Diagram 4-10. We excluded showing the other possible board configurations of Diagram
4-11 because they are symmetrical to the board configurations of Diagram 4-10. In each
potential Slice 1 or 3 of Diagram 4-16, one can see that only one of the alliance members
is able to move in his next turn. The other member will either have to move in the other
slice of the Slices 1 & 3 pair (if move is available) or move on Slice 2 or Slice 4. If an
alliance member is forced to play on Slice 2 or Slice 4, then that member must follow the
second part of the alliance’s winning strategy. That second part will be explained later in
this chapter.
The following explains the winning strategy for the alliance if the alliance were to play
on a potential Slice 1 or 3 that already has five pieces, with two of those pieces belonging
to M.
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The first thing the alliance will need to check is to see if both players can still move on
the same slice. These situations typically occur if M does not attempt to reserve spaces in
Slices 1 or 3, as indicated by the sample 4x3x4 game shown below.
A sample 4x3x4 game (L-M-R) R to play after 3.75 turns
M M •
• •• •
• ••
M M
M
M
•
• M •
• M ••
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-17 After M’s fourth turn (indicated by shaded square), both alliance members can move on Slice 3.
In Diagram 4-17, one can see that both R and L can move on Slice 3 in their next turns.
However, if both alliance members are unable to move on the same potential Slice 1 or 3,
then the first alliance member to move next on the slice takes the move, while the other
alliance member must move on a different slice. The following shows a sample game
where such a scenario can occur.
A sample 4x3x4 game (L-M-R) R to play after 3.75 turns
M M •
• •• •
• ••
M M
M
M
•
• M •
• M ••
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-18 After M’s fourth turn (indicated by shaded square), only one alliance member can move on Slice 3.
One can see that after R moves on Slice 3, L is no longer able to move on Slice 3. In this
particular example, L cannot move on Slice 1 either, so L must perform the second part
of the alliance’s winning strategy, which we will cover later.
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7) Alliance plays on a slice with six pieces
We will now examine the alliance’s winning strategy if their next playable slice already
has six pieces.
A sample 4x3x4 game (M-R-L) R to play after 4.5 turns
M • M •
M • •
•• ••
M M
M M M
•• ••
• M M •• •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-19 After M’s fifth turn (indicated by shaded square), only R can move on Slice 3.
Typically if an alliance member is able to move on a Slice 1 or 3 that has six pieces, then
M did not attempt to reserve spaces in that slice, which puts M in a severe disadvantage
against the alliance. In those instances, only one of the alliance members is able to move
on that slice because the six pieces (usually two L, two R, and two M pieces) occupy ten
spaces on a slice, leaving only two remaining spaces left. In Diagram 4-19, only R is
able to move on Slice 3, leaving L to play on another slice. In this particular example,
however, M is eliminated after R plays on Slice 3 as M has no more remaining moves to
play in this game.
We have demonstrated all possible scenarios of the alliance playing on a potential Slice 1
or 3. At some point during the game, the alliance member will not be able to play on
either Slice 1 or Slice 3 anymore. Then, the alliance member must follow the second part
of the alliance’s winning strategy, which we will cover in the next section.
4.2 – Part 2 of Alliance’s Winning Strategy
Earlier in this chapter, we mentioned that when an alliance member is no longer able to
move on either Slice 1 or 3, he must perform the second part of alliance’s winning
strategy against M. We continue the proof of Proposition 4-1 by explaining what that
second part is.
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•• ••
••
•• ••
The second part of the alliance’s winning strategy is similar to the second part of the
alliance’s winning strategy we covered in previous chapters:
Strategy: The alliance must cover the corresponding squares above and below any
remaining moves for M to block him from playing them. However, if the alliance is
unable to cooperatively block an available move for M, then both players should strive to
either block available M moves that only need one piece to block or partially block an
available M move.
There are two cases to consider when the alliance starts the second part of their winning
strategy. The first case is when both alliance members are unable to move on Slices 1
and 3 in their next turn. The second case is when only one alliance member is unable to
move on Slices 1 and 3 in his next turn. We will start the analysis of the winning strategy
with the first case.
Case 1: Both alliance members can no longer play on Slices 1 and 3
If both alliance members can no longer move on Slices 1 and 3 in their next turn, then
they should cooperatively block an available M space in their next turn if the opportunity
exists. The following diagram shows an example of this strategy.
A sample 4x3x4 M-L-R game M to play next after 5 turns
M • M •
M • •
• •
•
• •
• • M ••M
•
M • M
M •M
M •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-20 After M’s fifth turn (indicated by the shaded square), the alliance plays moves
above and below an available M move to block it (indicated by the x)
Diagram 4-20 shows a sample 4x3x4 M-L-R game where the alliance is no longer able to
play on Slices 1 and 3 after M’s fifth turn (indicated by the shaded square). At this point,
the alliance is forced to play on Slices 2 and 4, so their winning strategy is to
cooperatively block one of M’s remaining moves by playing pieces above and below an
available space. In Diagram 4-20, L and R have chosen the first moves from their
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•• •• ••
•••
•• •• •••
•••
respective ordered available moves sets that will block one of the two remaining M
moves on Slice 3 (indicated by the ‘x’).
However, simply choosing the first available L and R moves that cooperatively block M
is not an optimal strategy for the alliance. The following diagrams will demonstrate why.
A sample 4x3x4 game (L-R-M) M to play next after 4.5 turns
• •
• • M
• • M •
M
•••M
• M•
• • M •• •
•• M•
M •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-21 After M’s fourth turn (indicated by the shaded square), the alliance plays moves
above and below an available M move in Slice 3 (as indicated by the x).
In Diagram 4-21, the alliance cooperatively blocks one of M’s remaining moves on Slice
3 after M’s fourth turn. These moves were selected by picking the first moves from L and
R’s ordered available moves set that can block one of M’s available moves. Those
moves are not an optimal winning strategy for the alliance because there is another pair
of moves that the alliance can make that can block two of M’s available moves.
A sample 4x3x4 game (L-R-M) M to play next after 4.5 turns
• •
• • • M
• •M •
•••M
• M
• M•
• M •• •
••M•
M •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-22 After M’s fourth turn (indicated by the shaded square), the alliance plays
moves above and below an available M move in Slice 3 that also block an available M move in Slice 1 (indicated by the two x’s)
In Diagram 4-22, the alliance cooperatively moves on Slices 2 and 4 after M’s fourth turn
that not only blocks an available M move in Slice 3, but also blocks an M move in Slice
1.
Therefore, the alliance’s winning strategy for Case 1 is as follows.
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••
•••
••
•••
••
• If there are cooperative moves that can block more than one of M’s
available moves, then the alliance should play them.
• If that is not possible, then the alliance should play a pair of cooperative
moves in their next turns that can block one of M’s available moves.
• If cooperative blocking moves are not available, then both players should
block different available moves for M in their next turns.
If none of the above applies to an alliance member, then the member will need to
iterate through his available moves set until he finds a valid move. A valid move
in this case is one that would not prevent the other alliance member from playing
a piece that would have blocked one of M’s available moves.
Now that we have stated the alliance’s winning strategy for the case where both alliance
members cannot move on Slices 1 and 3 in their next turn, let’s examine the case where
only one of the alliance members cannot move on Slices 1 and 3 in his next turn.
Case 2: Only one alliance member can no longer play on Slices 1 and 3
The following diagram shows an example of what happens when one alliance member is
no longer able to play on Slices 1 and 3.
A sample 4x3x4 game (L-M-R) M to play next after 5.25 turns
M M •
• •
• M •
M •
•
M• •
• •
• •
M M •• •
M •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-23 After M’s fourth turn (indicated by the shaded square), R plays on Slice 3. L is unable to play on Slice 1 or Slice 3,
so he plays the first move that blocks M (indicated by the x).
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••
•• ••
••
•••
••
•• ••
•••••
•••
••
In Diagram 4-23, after M makes his fourth turn (indicated by the shaded square), only
one of the alliance members (R) is able to move on Slice 3. L would then need to play a
piece that would block one of M’s moves, which he does on Slice 2.
The one alliance member who cannot play on Slice 1 or Slice 3 should look for moves
that can block one of M’s available spaces using one piece. However, this alliance
member must be careful if he happens to go before the other alliance member who can
play on Slice 1 or 3. Here’s why.
A sample 4x3x4 game (M-R-L) R to play next after 4.5 turns
M • • •
M • •
• •
•
•
M •
• •
M M •
M M •
M • •
M M •
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-24 After M’s fifth turn (indicated by the shaded square), R should avoid
playing on the squares marked with ‘x’ as those spaces will be blocked by L’s next move on Slice 1 (indicated by the open squares).
Diagram 4-24 shows a sample 4x3x4 game with R going next. R could play on (2, 0, 1)
or (0, 1, 1) to block one of M’s remaining moves. However, L still has a move left on
Slice 1 (indicated by the open squares) that would block two of M’s spaces in Slice 2
(indicated by the x’s). Therefore, R’s move on Slice 2 would be a wasted move for the
alliance since R could have blocked another M move that would not have been blocked
by L’s move.
A sample 4x3x4 game (M-R-L) R to play next after 4.5 turns
M • • •
M • • •
• •
•
•
• M •
• •
M M •
M M
M
M M
Slice 1 Slice 2 Slice 3 Slice 4
Diagram 4-25 After M’s fifth turn (indicated by the shaded square), R played on Slice 4,
blocking one of M’s moves on Slice 3 (indicated by the x). L played on Slice 1, removing the last of M’s remaining moves.
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In Diagram 4-25, R ignores playing on squares marked with x’s in Diagram 4-24.
Instead, R plays on the upper-left corner of Slice 4, blocking M from playing on the
upper-left corner of Slice 3 (indicated by the x). L plays on Slice 1 after R, which
removes the last of M’s remaining moves, resulting in a loss for M.
The following summarizes the alliance’s winning strategy for Case 2.
If only one alliance member cannot play on Slice 1 or Slice 3 in his next turn, he
should play a move that can fully block an available M space using only one
piece. However, if that member goes first, he must not block an available M
space that will also be blocked by the other alliance member’s next move on a
slice that is in the Slices 1 & 3 pair.
This completes our analysis of the second part of the alliance’s winning strategy.
4.3 – Results from Simulation using Alliance’s Winning Strategy
Now that we have described the alliance’s winning strategy to prevent M from winning a
4x3x4 game (where M plays on the 4 component), we conclude the proof of Proposition
4-1 by proving that the winning strategy works. We have done this by creating an
application that simulates a three-player Domineering game and implemented the
winning strategy described in the last two sections for two of the players. The application
performs every possible move M can do throughout the game and chooses moves for L
and R based on the winning strategy.
There is one thing to be aware about this application—it will stop analyzing a game if M
is eliminated before L or R. The goal of the application is to check if there is a game
where M wins against the alliance, as that would disprove our winning strategy and
ultimately Proposition 4-1 (M is unable to force a win in a 4x3x4 game, where M plays
on the 4 component, if the L & R alliance teams up against him). Therefore, if M is
eliminated before L or R, then we do not care how that game ends.
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Our application generated six text files from the 4x3x4 game simulation, one for each
turn order (M-L-R, M-R-L, L-R-M, R-L-M, L-M-R, R-M-L). The following is an
excerpt from one of those text files:
Program started at: 1284406810 Dimensions of Domineering board: 4x3x4 Turn order for all games in this simulation: left, right, middle
Game Number: 1 + (10000000 x 0) Winner: left/right
left: 0,1,0 3,1,0 2,1,0 3,0,2 1,1,2 2,1,1 right: 0,0,0 2,0,0 0,0,2 2,2,2 1,0,3 2,1,3 middle: 1,1,0 1,2,0 0,1,1 0,2,1 2,0,1 # of Remaining Moves: - left: 5 - right: 5 - middle: 0
.
.
.
Game Number: 114377 + (10000000 x 0) Winner: left/right
left: 0,1,0 0,1,2 1,1,2 3,1,0 1,1,0 0,0,1 right: 0,0,0 0,0,2 2,0,2 2,0,0 2,1,1 0,2,1 middle: 3,2,2 3,1,2 2,2,2 2,1,2 2,2,0 # of Remaining Moves: - left: 5 - right: 7 - middle: 0
Program finished at: 1284406831
Diagram 4-26 An excerpt from the output of the 4x3x4 L-R-M game simulation.
The first lines of the output indicate the time the program started, the dimensions of the
board, and the turn order for all games simulated by the application. The simulation
stops analyzing a game when either M wins or has no more available moves (regardless
of whose turn it is). When that happens, it outputs how many times the simulation
stopped analyzing a game (indicated in the Game Number line), whether M won or not,
and what moves each player made for that specific game. It also writes the number of
remaining moves left for each player. When the application finishes traversing through
all possible games, it writes the time before it quits.
In each turn order output file, performing a search for “Winner: middle” yielded no
results. This is how we determine that there are no games where M won when the
alliance performed its winning strategy. If you were to take a random game from any of
the output files and played the moves indicated in that game, it will show that the alliance
performed their winning strategy to prevent M from winning.
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Although our output text files indicate that M did not win any games, there are certain
games where the last move recorded was made by M. To prove that M will not be able to
force a win, the application performs an additional move for both alliance members to
show that the alliance has at least one more move after M’s final move. There are some
games where one alliance member is eliminated before this additional move can be made.
However, the other alliance member is still able to move after M’s final move, which is a
loss for M. The “# of remaining moves” line in the output reflects the additional moves
made by the alliance after M’s final turn.
This concludes the proof of Proposition 4-1.
The following two charts compare the performance of our simulation using the alliance’s
winning strategy against a simulation that perform a turn-order sub-tree exhaustive search
of a 3D Domineering game.
4x3x4 3D Domineering Turn-Order Sub-tree Simulations Using Winning Strategy
Turn Order Duration Number of end nodes MLR 251 seconds 1,415,738 MRL 259 seconds 1,455,637 LMR 81 seconds 453,729 RML 83 seconds 458,325 LRM 21 seconds 114,377 RLM 21 seconds 116,839
Total: 716 seconds 4,014,645
Diagram 4-27
Exhaustive 3D Domineering Turn-Order Sub-tree Simulation
Size & Turn Order Duration Number of leaf nodes 3x3x3 - MLR Approx. 8 hours 1,034,224,512
Diagram 4-28
The chart in Diagram 4-27 demonstrates how long each of the six turn-order simulations
lasted, how many end nodes they reached, and the total sum of each simulation’s stats.
Diagram 4-28 lists the performance of the simulation that traverses the game tree of a
3x3x3 M-L-R game.
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In Chapter 1, we stated that in order to construct a game tree for a combinatorial game,
every possible move that could be made at each position by all players must be listed,
regardless of turn order. The simulation in Diagram 4-28 does not construct the entire
game tree for the 3D Domineering game. Instead, it only considered moves at each
position for whoever’s turn it was, which is why it is labeled as a turn-order sub-tree
simulation. This was done because calculating the complete game tree would have taken
the simulation a long time to compute. The total number of leaf nodes in complete 3x3x3
game tree is significantly higher than the number of leaf nodes in its turn-order sub-tree
counterpart. To demonstrate the size disparity between the complete tree and the turn-
order sub-tree, we will estimate the total number of leaf nodes in a complete 3x3x3 game
tree.
At each position of Diagram 4-28’s 3x3x3 turn-order sub-tree, the simulation only
considers the current player’s options instead of all three players’ options. In other
words, 2/3 of a complete 3x3x3 game tree’s branches are pruned at each level in the sub-
tree simulation. We can approximate the total number (K) of end nodes of Diagram 4-
28’s 3x3x3 simulation with the following equation:
K @�(2/3)n ∏�J
where J represents the total number of leaf nodes in a complete 3x3x3 game tree, and n
represents the height of the complete game tree. There are 27 spaces in a 3x3x3 game,
and if each player tried to pack as many pieces as possible in the board, there would be a
total of 27/2 = 13 moves made by all three players. Thus, we can presume that an upper
bound for the height of the 3x3x3 game tree is 13. If we were to rewrite the equation to
solve for J, we would get:
J @�(3/2)13 ∏��K ≈ 194.61 ∏��K
Replacing K with the number of leaf nodes defined in Diagram 4-28 for a 3x3x3 turn-
order sub-tree, the total number of leaf nodes in a complete 3x3x3 game tree is
approximately 201,280,264,483. This is significantly more than the leaf nodes listed for
the 3x3x3 turn-order sub-tree in Diagram 4-28. We can make the case that the total
number of leaf nodes for a complete 4x3x4 game tree would be significantly higher than
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the number of leaf nodes in a complete 3x3x3 game. We make this point because the
total time and number of end nodes from our simulation would clearly be significantly
less than an analysis of a complete 4x3x4 game.
Using our application, we have demonstrated that regardless of turn-order, M cannot
force a win in a 4x3x4 Domineering game, where M plays on the second 4 component, if
the alliance teams up against M using the winning strategy we outlined in this chapter.
We have demonstrated in the last two chapters that M cannot force a win in a 4x4x3
game if L and R collude against him, regardless of turn order or whether M plays on the 3
component or the second 4 component. If M cannot force a win if L and R team up
against him, it also means that L cannot force a win if R and M team up against him and
R cannot force a win if L and M team up against him. Therefore, no player can force a
win in a 4x4x3 game, regardless of board orientation or turn order. Thus, we have
proven that a 4x4x3 game is solved to be in the outcome class .
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5.0 – Future Work
Completely solving a 4x4x3 Domineering game using the strategies outlined in the
previous chapters opens the possibility of solving 3D Domineering boards of larger sizes
without having to do a complete traversal of their game trees. We previously determined
that the alliance’s optimal strategy to prevent M from winning 3x3x3 and 4x4x3 (where
M plays on the 3 component) games is to cover as much of the middle slice as possible.
One consideration for future work is to investigate if there is a limit to how large a game
of m x n x 3 (where m > 4, n > 4, and M plays on the 3 component) can be where
blocking the middle slice remains an effective strategy for the alliance to block M.
Another consideration for future work is to test if having the alliance play on alternating
slices is an effective strategy to prevent M from winning regardless of how many slices
M can play on. We found that when M plays on four slices in a 4x3x4 game, the strategy
to block M is to play on alternating slices (Slices 1 & 3). If M played on five slices (for
example, 5x5x5), can the alliance prevent M from winning by playing on Slices 2 & 4?
And if M played on six slices (for example, 6x6x6), can the alliance prevent M from
winning if they played on Slices 1, 3, and 5? What if the board size was skewed to M’s
favor, such as a 3x3x8 board, with M playing on the 8 component? Would playing on
alternate slices still be enough for the alliance to prevent M from winning? How big does
the board have to skew in M’s favor before M can force a win despite the other two
players teaming up against him? These are questions that can be explored with further
research.
Yet another consideration for future work is improving how the simulation application
chooses moves for the alliance. For the 4x3x4 game, the application either chose hard-
coded moves or the first available moves that fulfilled certain criteria (blocked M’s move,
did not block other alliance member, etc). An improvement to the application would
consider ranking available moves for both alliance members and choosing the one that
would most help the alliance defeat M. The process of ranking moves would prolong the
amount of time it takes for the application to simulate a three-player game. However, its
implementation could be applied to larger boards without having to write specialized
code for the game similarly to the ones written for the 4x3x4 game.
110
References
[ANW_07] Albert, Michael H., Richard J. Nowakowski, and David Wolfe. Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters, Ltd., Massachusetts, 2007.
[Con_76] Conway, J.H. On Numbers and Games. Academic Press, London, 1976.
[Bou_02] Bouton, C.L. “Nim, a Game with a Complete Mathematical Theory.” Annals of Mathematics. 3-2:35-39, 1902.
[Ros_03] Rosen, Kenneth H. Discrete Mathematics and Its Applications, 5th
Edition. McGraw Hill, New York, 2003.
[BUH_00] Breuker, D.M., J.W.H.M. Uiterwijk, H.J. van den Herik. “Solving 8 x 8 Domineering.” Theoretical Computer Science. 230:195-206, 2000.
[Bul_02] Bullock, Nathan. “Domineering: Solving Large Combinatorial Search Spaces.” University of Alberta, 2002.
[Str_85] Straffin, Jr., Philip D. “Three Person Winner-Take-All Games with McCarthy’s Revenge Rule.” The College Mathematics Journal. 16:386-394, November 1985.
[Pro_00] Propp, James. “Three-Player Impartial Games.” Theoretical Computer Science. 233:263-278, 2000.
[Cin_08] Cincotti, Alessandro. “Three-Player Domineering.” Proceedings of World Academy of Science, Engineering, and Technology 36:92-95, December 2008.
[Cin_05] Cincotti, Alessandro. “Three-Player Partizan Games.” Theoretical Computer Science. 332:367-389, 2005.
[Kla_10] Klassen, Matt. Personal communication, 2010.
[Cin_09] Cincotti, Alessandro. “Further Results on Three-Player Domineering.” Proceedings of World Academy of Science, Engineering, and Technology 39:194-196, March 2009.
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